Sébastien Breteaux

A GEOMETRIC DERIVATION OF THE LINEAR BOLTZMANN EQUATION FOR A PARTICLE INTERACTING WITH A GAUSSIAN RANDOM FIELD

Keywords: 2000 Mathematics Subject Classification. 82C10, (60K37, 81E, 81S, 81D30, 82B44, 82C40) Linear Boltzmann equation, Processes in random environments, Quantum field theory, Coherent states, Kinetic theory of gases

In this article the linear Boltzmann equation is derived for a particle interacting with a Gaussian random field, in the weak coupling limit, with renewal in time of the random field. The initial data can be chosen arbitrarily. The proof is geometric and involves coherent states and semi-classical calculus.

Introduction

In this article we derive the linear Boltzmann equation for a particle interacting with a translation invariant centered Gaussian random field. The evolution of this particle is described by the Liouville -Von Neumann equation with a Hamiltonian -∆ x + V h ω (x), where the potential depends on a random parameter ω. In the weak coupling limit, the dependence of the random potential with respect to h is V h ω = √ hV ω , where h represents the ratio between the microscopic and macroscopic scales. We consider the limit h → 0. In the case of a Gaussian random field the weak coupling limit and the low density limit agree. Through an isomorphism between the Gaussian space L 2 (Ω P , P; C) associated with L 2 (R d ; R) and the symmetric Fock space ΓL 2 (R d ) associated with L 2 (R d ; C), multiplication by V ω (x) corresponds to the field operator √ 2Φ(V (x -•)) for some function V . We can thus express the Hamiltonian in the Fock space and approximate the dynamics by an explicitly solvable one whose solutions are coherent states. The geometric idea behind the computations is due to the fact that the initial state is the vacuum, and we can thus expect that for short times the system is approximately in a coherent state whose parameter moves slightly in the phase space. This parameter in the (infinite dimensional) phase space then gives the important information in the limit h → 0. The computations done with this solution allow us to recover the dual linear Boltzmann equation for short times for the observables. A renewal of the random field allows us to reach long times.

The derivation of the linear Boltzmann equation has been studied for both classical and quantum microscopic models. In the classical case Gallavotti [START_REF] Gallavotti | Divergences and the approach to equilibrium in the lorentz and the wind-tree models[END_REF] provided a derivation of the linear Boltzmann equation for Green functions in the case of a Lorentz gas. Later Spohn [START_REF] Spohn | Kinetic equations from hamiltonian dynamics: Markovian limits[END_REF] presented a review of different classical microscopic models and of kinetic equations obtained as limits of these models, with emphasis on the approximate Markovian behaviour of the microscopic dynamics (some quantum models were also studied). Boldrighini, Bunimovich and Sinaȋ [START_REF] Boldrighini | On the Boltzmann equation for the Lorentz gas[END_REF] gave a derivation of the linear Boltzmann equation for the density of particles in the case of the Lorentz model. In the quantum case, Spohn derived in [START_REF] Spohn | Derivation of the transport equation for electrons moving through random impurities[END_REF] the radiative transport equation in the spatially homogenous case. Later Ho, Landau and Wilkins studied in [START_REF] Ho | On the weak coupling limit for a Fermi gas in a random potential[END_REF] the weak coupling limit of a Fermi gas in a translation invariant Gaussian potential (and other random potentials). Their proofs made use of combinatorics and graph techniques. In the case of a particle interacting with a Gaussian random field (the setting of this article) Erdős and Yau [START_REF] Erdös | Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation[END_REF] removed the small time restriction, and also generalized the initial data to WKB states, using methods with graph expansions. Developements of that method by Chen [START_REF] Chen | Localization lengths and Boltzmann limit for the Anderson model at small disorders in dimension 3[END_REF] and Erdős, Salmhofer and Yau [START_REF] Erdős | Quantum diffusion of the random Schrödinger evolution in the scaling limit[END_REF][START_REF] Erdős | Quantum diffusion of the random Schrödinger evolution in the scaling limit. II. The recollision diagrams[END_REF] did not require a Gaussian form for the random field but still supposed an initial state of the WKB form. The linear Boltzmann equation was derived in the radiative transport limit by Bal, Papanicolaou and Ryzhik [START_REF] Bal | Radiative transport limit for the random Schrödinger equation[END_REF] in the quantum case, and by Poupaud and Vasseur [START_REF] Poupaud | Classical and quantum transport in random media[END_REF] in the classical case using a potential stochastic in time. This assumption automatically ensures that there is no self-correlation in the paths of the particles and simplifies the problem. Later Bechouche, Poupaud and Soler [START_REF] Bechouche | Quantum transport and Boltzmann operators[END_REF] used similar techniques to get a model for collisions at the quantum level and obtain a kind of quantum linear Boltzmann equation. For these stochastic methods the initial state can be arbitrary but the potential is almost surely bounded, which excludes Gaussian or Poissonian random fields.

Remarks. Our derivation is given in the case of a Gaussian random field but other random fields could be considered with the same type of methods, for example a Poissonian random field. Note that the weak coupling and low density limit do not then agree.

Our approach allows initial states to be arbitrary, contrary to WKB initial states. The framework of quantum field theory allows to see how geometry in phase space is involved. We use the viewpoint of Ammari and Nier [START_REF] Ammari | Mean field limit for bosons and infinite dimensional phasespace analysis[END_REF] but in a case that is not in the framework chosen by the authors. Indeed we are not dealing with a mean field limit and the introduction of a parameter ε is an artifact that allows us to keep track of the importance of the different terms. We thus adopt a different viewpoint from the graph expansions or the stochastic viewpoint adopted in other works on the subject, and this allows us to keep track of the geometry.

However, we cannot as of yet reach times of order 1 like in [START_REF] Erdös | Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation[END_REF][START_REF] Chen | Localization lengths and Boltzmann limit for the Anderson model at small disorders in dimension 3[END_REF][START_REF] Erdős | Quantum diffusion of the random Schrödinger evolution in the scaling limit. II. The recollision diagrams[END_REF][START_REF] Erdős | Quantum diffusion of the random Schrödinger evolution in the scaling limit[END_REF]. As we do not get the approximate Markovian behaviour in a satisfying way, we need to introduce a renewal of the random potential. Attal and Pautrat in [START_REF] Attal | From repeated to continuous quantum interactions[END_REF] and Attal and Joye in [START_REF] Attal | Weak coupling and continuous limits for repeated quantum interactions[END_REF] deal in a more sophisticated way with interactions defined piecewise in time. Other Ansätze may give a better approximation of the solution to the initial problem and give the Markovian behaviour of the evolution.

One of the important tools in our derivation of the linear Boltzmann equation is the use of a priori estimates to show that we do not lose too much mass in the measures during our approximations. The mass conservation and positivity properties of the linear Boltzmann equation then allow us to complete the proof.

Our result holds in dimension d ≥ 3 as dispersion inequalities for the free Schrödinger group provide the time integrability needed for some expressions. It may be possible to reach the limit case of dimension d = 2.

Outline of the article. In Section 2, we describe the quantum model, state the main result and give the structure of the proof. We then recall some facts about the linear Boltzmann equation in Section 3. We specify the link between the Gaussian random field and the symmetric Fock space in Section 4 and thus obtain a new expression for the dynamics. We study an approximate dynamics in Section 5. We use this explicit solution to compute the measurement of an observable for short times in Section 6. We control the error involved in this approximation in Section 7. And finally, we combine these results to complete the proof in Section 8.

Model and result

2.1. The model. Let ω ∈ Ω P be a random parameter and x ∈ R d (d ≥ 1) a space parameter. Let V h ω (x) the translation invariant centered Gaussian random field with mean zero and covariance hG

(x -x ′ ), such that Ĝ = | V | 2 with V ∈ S R d ; R . We consider the Liouville -Von Neumann equation (2.1) ih∂ t ρ t,ω = [H h ω , ρ t,ω ] , H h ω = -∆ x + V h ω (x) , with an initial condition ρ h 0,ω = ρ h 0 in the set of states on L 2 x = L 2 (R d x ; C) (i.
e. the subset of the non-negative trace class operators L + 1 (L 2 x ) whose trace is 1). Note that [A, B] denotes the commutator AB -BA of two operators.

We now introduce the renewal of the random field. We fix a time T , an integer N and set ∆t = T /N . For a state ρ on L 2

x , let

G h t (ρ) = ˆe-i t h H h,ω ρ e i t h H h,ω dP(ω) , (2.2)
ρ h t = G h t (ρ) , (2.3) ρ h N,∆t = (G h ∆t ) N (ρ) . (2.4)
With t k = k∆t, the dynamics is defined piecewise on the intervals [t k-1 , t k ] by the Hamiltonians H h,ω k = -∆ x + V h,ω k (x) with independent random fields V h,ω k , ω k in copies of Ω P . Thus we get, for an initial data ρ 0 ∈ L 1 + (L 2 x ), that the system is in the state ρ h N,∆t at time T .

The main result

. Let b ∈ C ∞ 0 (R 2d x,ξ ). The measure of the observable b W (hx, D x ) in a state ρ on L 2
x is given by

m h (b, ρ) = Tr b W (hx, D x )ρ ,
where the Weyl quantization (see for example Martinez's book [START_REF] Martinez | An introduction to semiclassical and microlocal analysis[END_REF]) is defined by

b W (hx, D x )u (x) = (2π) -d ˆR2d x ′ ,ξ e i(x-x ′ ).ξ b h x+x ′ 2 , ξ u(x ′ ) dx ′ dξ .
Semiclassical measures (and microlocal defect measures) have been studied by, among others, Gérard [START_REF] Gérard | Mesures semi-classiques et ondes de Bloch[END_REF][START_REF] Gérard | Microlocal defect measures[END_REF], Burq [START_REF] Burq | Mesures semi-classiques et mesures de défaut[END_REF], Gérard, Markowich, Mauser and Poupaud [START_REF] Gérard | Homogenization limits and Wigner transforms[END_REF][START_REF] Gérard | Erratum: "Homogenization limits and Wigner transforms[END_REF] and Lions and Paul [START_REF] Lions | Sur les mesures de Wigner[END_REF]. Let us quote Theorem 2.1, which is a direct consequence of a theorem which can be found in [START_REF] Burq | Mesures semi-classiques et mesures de défaut[END_REF] (with

(ρ h ) replacing (|u k u k |) for weakly convergent sequence (u k ) of L 2 x ).
Theorem 2.1. Let (ρ h ) h∈(0,h0] , h 0 > 0 be a family of states on L 2

x . There exist a sequence h k → 0 and a non-negative measure µ on R 2d

x,ξ such that

∀b ∈ C ∞ 0 (R 2d x,ξ ) , lim n→+∞ m h k (b, ρ h k ) = ˆR2d x,ξ b dµ .
The measure µ is called a semiclassical measure (or Wigner measure) associated with the sequence (ρ h k ). Let M(ρ h , h ∈ (0, h 0 ]) be the set of such measures. If this set is a singleton {µ} then the family (ρ h ) is said to be pure and associated with µ.

By a simple sequence extraction out of the range of the parameter h, the family can always be assumed to be pure. For evolution problems the fact that the sequence extraction can be performed uniformly for all times is a property to be proved.

We can now state the main theorem of this article.

Theorem 2.2. Assume d ≥ 3. Let ∆t = h α , N = N h = T /h α , α ∈ ( 3 4 , 1). Assume that (ρ h ) h∈(0,h0] is pure and associated with µ 0 such that µ 0 (R d x ×R d * ξ ) = 1.
Then (ρ h N,∆t ) h∈(0,h0] is pure and associated with µ T , where (µ t ) t solves the linear Boltzmann equation From these statements, the lower bound

(2.5) ∂ t µ t (x, ξ) + 2ξ.∂ x µ t (x, ξ) = ˆσ(ξ, ξ ′ ) δ |ξ| 2 -|ξ ′ | 2 (µ t (x, ξ ′ ) -µ t (x, ξ)) dξ ′ with the initial condition µ t=0 = µ 0 and σ(ξ, ξ ′ ) = 2π| V (ξ -ξ ′ )| 2 . The Fourier transform on R d is here û(ξ) = F u(ξ) = ´Rd x e -ix.ξ u(x) dx. Sketch of the Proof. Let µ T in M(ρ h N,∆t , h ∈ (0, h 0 ]).
ˆRd x ×R d * ξ b dµ T ≥ ˆRd x ×R d * ξ b d(B(T )µ 0 )
follows. Since this inequality holds for any non-negative b from the set of smooth functions with compact support

C ∞ 0 (R d x × R d * ξ ), which is dense in the set of contin- uous functions vanishing at "infinity" C 0 ∞ (R d x × R d * ξ ), whose dual is the set of Radon measures M b (R d x × R d * ξ ), we get µ T | R d x ×R d * ξ ≥ B(T )µ 0 | R d x ×R d * ξ .
But we also have

B(T )µ 0 (R d x × R d * ξ ) =
µ T R d x × R d * ξ = 1 , µ T R d x × {0} ξ = 0
and µ T = B(T )µ 0 . Hence the result.

Remark 2.3.
Step 2 is the technical part and requires various estimates developed in this article.

Remark 2.4. Let us justify the scaling in the Weyl quantization. Physically the parameter h is the quotient of the microscopic scale over the macroscopic scale, either in time or in position. Thus if we consider an observable b(X, Ξ) varying on a macroscopic scale, the corresponding observable on the microscopic scale will be b(hx, ξ).

The scaling of the random field according to the covariance hG(xx ′ ) is done on a mesoscopic scale imposed by the kinetic regime. In microscopic variables, consider a particle moving among obstacles with a velocity v ∝ 1 and a distance of interaction R ∝ 1. During a time T the particle sweeps a volume of order vT R d-1 . In the kinetic regime it is assumed that during a long microscopic time T = t/h with t ∝ 1 the macroscopic time, the average particle encounters a number ∝ 1 of obstacles. We denote by ρ the density of obstacles and thus obtain ρ = 1/vT R d-1 ∝ h. To get this density of obstacles we need the distance between two nearest obstacles to be of order h -1/d .

Thus we consider a Schrödinger equation of the form

i∂ T ψ = -∆ x ψ + V h ω (x) ψ ,
that is,

ih∂ t ψ = -∆ x ψ + V h ω (x) ψ . A translation invariant Gaussian random field of covariance G(x -x ′ ), Ĝ = | V | 2 , is of the form V * W ω
, where W ω is the spatial white noise and V describes the interaction potential. In the kinetic regime the obstacles are spread at the mesoscopic scale h 1/d . Only the white noise W h ω is rescaled (and not V ) according to

∀ϕ ∈ S(R d ; R) , ˆϕ(h 1/d x) W h ω (x) dx = ˆϕ(x) W ω (x) dx ,
i.e., W h ω (x) = hW ω (h 1/d x). Thus we get V h ω = hV * W ω (h 1/d •) and G h = hG. To prove Theorem 2.2 we first consider the case without the renewal of the stochastics, i.e., N = 1 for short times in Sections 5, 6, 7 and then glue together the estimates obtained this way N times for N "big" in Section 8. To simplify the problem of finding estimates for short times we approximate the equation by a simpler one which is solved and studied in Section 5. In Section 6, using the solution to the approximated equation, we carry out explicit computations which give rise to the different terms of the dual linear Boltzmann equation. Then we control the error between the solutions of the approximated equation and the exact equation in Section 7. All these computations are done within the framework of quantum field theory. This allows us

• to use conveniently the geometric content of coherent states, • to keep track of the different orders of importance of the different terms by using the Wick quantization with a parameter ε.

We expose the correspondence between the stochastic and Fock space viewpoints in Section 4.

Remark 2.5. Our initial data (ρ h ) h∈(0,h0] are assumed to belong to L + 1 L 2 x with Tr ρ h = 1. We thus make estimates for states ρ in L + 1 L 2 x , with Tr ρ = 1 with constants independent of ρ.

The linear Boltzmann equation

Information on the linear Boltzmann equation can be found in the books of Dautray and Lions [START_REF] Dautray | Analyse mathématique et calcul numérique pour les sciences et les techniques[END_REF][START_REF] Dautray | Analyse mathématique et calcul numérique pour les sciences et les techniques[END_REF] or Reed and Simon [START_REF] Reed | Methods of modern mathematical physics[END_REF].

In Section 3 the set of values of functions is R when nothing is precised. We assume that σ ∈ C ∞ (R d ξ × R d ξ ′ ) and σ ≥ 0.

3.1. Formal definition. Since all the objects we use are diagonal in |ξ|, the following notations are convenient. Notation: Let 0 < r < r ′ < +∞, we define the Sobolev spaces

H n [r, r ′ ] = H n (R d x × A ξ [r, r ′ ]) where A ξ [r, r ′ ] is the annulus {ξ ∈ R d , |ξ| ∈ (r, r ′ )} in the variable ξ. When there is no ambiguity we write A ξ for A ξ [r, r ′ ]. We also write L 2 [r, r ′ ] for H 0 [r, r ′ ].
Definition 3.1. The linear Boltzmann equation is formally the equation, with initial condition µ t=0 = µ 0 ,

∂ t µ = {µ, |ξ| 2 } + Qµ ,
where the collision operator

Q is defined for b ∈ L 2 [r, r ′ ] by (3.1) Qb = Q + b -Q -b , with Q + b(x, ξ) = ˆRd ξ ′ b(x, ξ ′ ) σ(ξ, ξ ′ ) δ |ξ| 2 -|ξ ′ | 2 dξ ′ , Q -b(x, ξ) = b(x, ξ) ˆRd ξ ′ σ(ξ, ξ ′ ) δ |ξ| 2 -|ξ ′ | 2 dξ ′ .
The dual linear Boltzmann equation with initial condition b t=0 = b 0 is

∂ t b = -
Q L(H n [r,r ′ ]) ≤ C d sup |α|≤n ∂ α σ ∞,A 2 ξ [r,r ′ ] .
The group of space-translations (e 2tξ.∂x ) t preserves H n [r, r ′ ].

• 2ξ.∂ x +Q generates a semigroup (B T (t)) t≥0 bounded by exp(t Q L(L 2 [r,r ′ ]) ) since Q is bounded on L 2 [r, r ′ ].
The strongly continuous group (B T (t)) t≥0 preserves (1) the Sobolev spaces H n [r, r ′ ], for n ∈ N, (2) the set of functions with compact support, (3) the set of infinitely differentiable functions with compact support in preserves the non-negative functions for t ≥ 0 because Q + does, e -t n Qpreserves the non-negative functions as a multiplication operator by a positive function.

R d x × A ξ [r, r ′ ], C ∞ 0 (R d x × A ξ [r, r ′ ]), (
C ∞ 0 (R d x × A ξ [r, r ′ ]) = ∞ n=0 H n [r, r ′ ] {f, Supp f compact} . Point (
Since C ∞ 0 (R d x × A ξ ) ⊂ D(2ξ.∂ x )
we can give the following result.

Proposition 3.4. For all b 0 ∈ C ∞ 0 (R d x × A ξ ), b t = B T (t)b 0 is the unique solution in C 1 (R + ; L 2 [r, r ′ ]) ∩ C 0 (R + ; D(2ξ.∂ x )) to the Dual linear Boltzmann equation such that b t=0 = b 0 . Moreover ∀t ∈ R, b t ∈ C ∞ 0 (R d x × A ξ ). If b 0 is non-negative, then ∀t ≥ 0, b t is non-negative.

3.3.

The linear Boltzmann equation. The continuous functions vanishing at infinity and the Radon measures on a locally compact, Hausdorff space X are denoted by

C 0 ∞ (X) = {f ∈ C 0 (X) , ∀ε > 0 , ∃K compact s.t. ∀x / ∈ K , |f (x)| < ε} , M b (X) = (C 0 ∞ (X)) ′ . Proposition 3.5. The semigroup (B T (t)) t≥0 defined on C ∞ 0 (R d x ×R d * ξ ) extends to a strongly continuous group on (C 0 ∞ (R d x × R d * ξ ), • ∞ ) and defines by duality a (weak * continuous) group B(t) on M b (R d x × R d * ξ ).
Proof. Using a partition of the unity,

B T (t) extends to C ∞ (R d x ×R d * ξ ). Since B T (t) is positive, we have B T (t)( b ∞ ±b) ≥ 0 for all b in C ∞ 0 (R d x ×R d * ξ ) and so B T (t) b ∞ ≤ b ∞ . The group B T (t) thus extends continuously to C 0 ∞ (R d x × R d * ξ ). Definition 3.6. The linear Boltzmann group (B(t)) is defined on M b (R d x × R d * ξ ) by duality: let µ ∈ M b (R d x × R d * ξ ), then, for any t ∈ R, ∀b ∈ C 0 ∞ (R d x × R d * ξ ), B(t)µ, b = µ, B T (t)b .

3.4.

A Trotter-type approximation. This Section provides a result in the spirit of Trotter's approximation (e A/N e B/N ) N → e A+B useful to deal with the renewal of the stochasticity.

Proposition 3.7. Let b ∈ C ∞ 0 (R 2d x,ξ ), T > 0 and n ∈ N. There are constants C n,Q and C T,b such that for all N ∈ N * N n e T (2ξ.∂x+Q) b -e T N Q e T N 2ξ.∂x N b ≤ e T (2n+Cn,Q) C T,b T 2 N .
where,

for n ∈ N, N n (b) := sup |α|≤n ∂ α b ∞ . Notation 3.8. Let Q t = e t2ξ.∂x Qe -t2ξ.∂x ∈ L(L 2 [r, r ′ ]), i.e. Q t = Q +,t -Q -with Q +,t b(x, ξ) = ˆRd ξ ′ σ(ξ, ξ ′ ) δ |ξ| 2 -|ξ ′ | 2 b(x -2t(ξ ′ -ξ), ξ ′ ) dξ ′ .
Let also Q -,t = Q -to have consistent notations in the sequel. Let G Q (t, t 0 ) be the dynamical system associated with the one parameter family

(Q t ) in C(R; L(L 2 [r, r ′ ])) given by ∂ t b t = Q t b t b t=t0 = b 0 ∈ L 2 x,ξ , b t = G Q (t, t 0 ) b 0 . Note the relation B T (t) = G Q (t, 0)e 2tξ.∂x = e 2tξ.∂x G Q (0, -t). For b ∈ C ∞ 0 (R d x × A ξ [r, r ′ ]), let N n (Q) = sup b =0 N n (Qb) N n b and N n+1,n (s, Q -Q s ) = sup b =0 N n (Q -Q s )b s (1 + 2 |s|) n N n+1 b . Lemma 3.9. For any n ∈ N, s ≥ 0 and b ∈ C ∞ 0 (R d x × A ξ [r, r ′ ]
), there exist constants C 1 , and C 2 depending on d, r and r ′ such that

(1) N n (Q) ≤ C 1 , (2) N n+1,n (t, Q -Q t ) ≤ C 2 , ( 3
) N n (e 2tξ.∂x b) ≤ (1 + 2 |t|) n N n (b).
Proof. The first point is clear from the integral expression of Qb.

For the second point differentiate and estimate the integral formula for b

(x -2tξ, ξ)- b (x, ξ), with |α| ≤ n, ∂ α b(x -2tξ, ξ) -b(x, ξ) ≤ ˆt 0 ∂ α 2ξ.∂ x b(x -2sξ, ξ) ds ≤ 2 |ξ| t (1 + 2t) n N n+1 (b) .
The last point results from e 2tξ.∂x b (x, ξ) = b(x + 2tξ, ξ).

Lemma 3.10. Let b, b ∈ C ∞ 0 R d x × A ξ [r, r ′ ] , then for all t ≥ 0, e tQ b -G Q (t, 0)b = e tQ ( b -b) + ˆt 0 e (t-s)Q (Q -Q s )G Q (s, 0)b ds
and we have the estimate

N n e tQ b -G Q (t, 0)b ≤ e tNnQ N n ( b -b) + t 2 (1 + 2t) n e tNnQ sup s∈[0,t] N n+1,n (s, Q -Q s ) N n+1 G Q (s, 0) N n+1 (b) .
Proof. The equality is clear once we have computed that both sides satisfy the equation

∂ t ∆ t = Q∆ t + (Q -Q t )G Q (t, 0)b .
The inequality then follows from Lemma 3.9.

Proof of Proposition 3.7. We fix N and forget the N 's in the notations concerning b. We set b t = B T (t) b and define bt piecewise on [0, T ] by setting

t k = kT N , bt k = e T N Q e T N 2ξ.∂x k b 0 and, for t ∈ [t k , t k+1 ), bt = e (t-t k )Q e (t-t k )2ξ.∂x bt k . Let δ k = N n b t k -bt k ; we get e T N Q e T N 2ξ.∂x bt k -e T N (2ξ.∂x+Q) b t k = e T N Q e T N 2ξ.∂x bt k -G Q T N , 0 e T N 2ξ
.∂x b t k and we can then use Lemma 3.10 to obtain

δ k+1 ≤ e T N NnQ 1 + 2 T N n δ k + T N 2 1 + 2 T N n e T N NnQ sup s∈[t k ,t k+1 ] N n+1,n s -t k , Q -Q s-t k sup s∈[t k ,t k+1 ] N n+1 G Q (s -t k , 0) e T N 2ξ.∂x b t k ≤ e T N NnQ e 2 nT N δ k + T N 2 e
T N NnQ C N,T where we introduced

C N,T,b = 1 + 2 T N n sup s∈[0,T /N ] N n+1,n (s, Q -Q s ) sup k∈{0,...,N -1} sup s∈[0,T /N ] N n+1 G Q (s, 0) e -T N Q b t k+1 .
Then we get the recursive formula

δ k+1 ≤ e T N (2n+NnQ) δ k + C N,T,b T N 2 e T N NnQ so that δ N ≤ e T (2n+NnQ) C N,T,b T 2 N . The only thing remaining is to observe that C N,T,b ≤ C T,b , with C T,b := (1 + 2T ) n sup s∈[0,T ] N n+1,n (s, Q -Q s ) sup sj ∈[0,T ] N n+1 G Q (s 1 , 0) e -s2Q b s3
and for a fixed T this quantity C T,b is finite, so that we get the result.

4.

From stochastics to the Fock space 4.1. The second quantization. The method of second quantization is exposed in the books of Berezin [START_REF] Feliks | The method of second quantization[END_REF] and Bratteli and Robinson [START_REF] Bratteli | Operator algebras and quantum statistical mechanics[END_REF], an introduction to quantum field theory and second quantization can be found in the book of Folland [START_REF] Folland | of Mathematical Surveys and Monographs[END_REF]. The series of articles of Ginibre and Velo [START_REF] Ginibre | The classical field limit of scattering theory for nonrelativistic many-boson systems[END_REF][START_REF] Ginibre | The classical field limit of scattering theory for nonrelativistic many-boson systems[END_REF][START_REF] Ginibre | The classical field limit of nonrelativistic bosons. I. Borel summability for bounded potentials[END_REF][START_REF] Ginibre | The classical field limit of nonrelativistic bosons. II. Asymptotic expansions for general potentials[END_REF] uses this framework with a small parameter to handle classical or mean field limits by extending the Hepp method [START_REF] Hepp | The classical limit for quantum mechanical correlation functions[END_REF]. We use the notation and framework of articles of Ammari and Nier [START_REF] Ammari | Mean field limit for bosons and infinite dimensional phasespace analysis[END_REF][START_REF] Ammari | Mean field limit for bosons and propagation of Wigner measures[END_REF] to handle the second quantization with a small parameter. For the convenience of the reader we expose briefly this framework.

Most of the operators on the Fock space in this article arise as Wick quantizations of polynomials. Definition 4.1. Let (H, •, • ) be a complex separable Hilbert space (the scalar product is C-antilinear with respect to the left variable). The symmetric tensor product is denoted by ∨. The polynomials with variable in H are the finite linear combinations of monomials Q : H → C of the form

Q (z) = z ∨q , Qz ∨p
where p, q ∈ N, Q ∈ L(H ∨p , H ∨q ) and •, • denotes the scalar product on H ∨q . The set of such polynomials is denoted by P(H).

The symmetric Fock space associated to H is

ΓH = ∞ n=0 Γ n H
with Γ n H = H ∨n the Hilbert completed n-th symmetric power of H and the sum is completed, the set of finite particle vectors Γ F H is defined as the Fock space but with an algebraic sum.

Let ε > 0. The Wick quantization of a polynomial is defined as the linear combination of the Wick quantizations of its monomials, and for a monomial Q we define Q W ick : Γ F H → Γ F H as the linear operator which vanishes on H ∨n for n < p and for n ≥ 0

Q W ick H ∨n+p = √ (n+p)!(n+q)! n! ε p+q 2 ( Q ∨ Id H ∨n ) ∈ L H ∨n+p , H ∨n+q . The field operator Φ ε (f ) (f ∈ H) is the closure of the essentially self-adjoint op- erator ( z, f + f, z ) W ick / √ 2. Using the Weyl operator W (f ) = exp(iΦ ε (f )) the coherent state E(f ) = W √ 2
iε f Ω can be defined, where Ω = (1, 0, 0, . . . ) ∈ ΓH is the empty state. The Weyl operators satisfy the relation

W (f ) W (g) = e -iε 2 ℑ f,g W (f + g) .
The second quantization dΓ ε (A) of a self-adjoint operator A on H is

dΓ ε (A)| D(A) ∨n,alg = ε (A ⊗ Id H ⊗ • • • ⊗ Id H + • • • + Id H ⊗ • • • ⊗ Id H ⊗ A)
and for a unitary U on H, the unitary operator Γ(U ) on ΓH is defined by

Γ(U )| H ∨n = U ∨n = U ⊗ • • • ⊗ U and thus Γ(e itA ) = exp it ε dΓ ε (A) . 4.2.
The expression of the dynamic in the Fock space. The relation between Gaussian random processes and the Fock space is treated in the books of Simon [START_REF] Simon | The P (φ) 2 Euclidean (quantum) field theory[END_REF] and Glimm and Jaffe [START_REF] Glimm | Quantum physics[END_REF], we recall a theorem about this relation. Theorem 4.2. Let V h (x) be the centered, translation invariant, gaussian random field with covariance hG(xy)

such that Ĝ = | V | 2 for some V ∈ S(R d ; R). The symmetric Fock space ΓL 2 (R d ; C) is unitarily equivalent to L 2 (Ω P , P; C) under a unitary D : ΓH C → L 2 (Ω P , P; C) such that • D Ω = 1, • D √ 2hΦ 1 (τ x V ) D -1 = V h (x), with V h (x)
seen as a multiplication operator on L 2 (Ω P , P; C).

For Hilbert spaces H and H ′ , Tr H ′ [A] denotes the partial trace of an operator

A ∈ L 1 (H ⊗ H ′ ), Tr H [Tr H ′ [A]B] = Tr H⊗H ′ [A(B ⊗ I H ′ )], ∀B ∈ on H ⊗ H ′ . Proposition 4.3. Let H h = -∆ x + √ 2hΦ 1 (τ x V ), with τ x f (y) = f (y -x) for x ∈ R d and f ∈ L 2 y . Then G h t (ρ) = Tr ΓL 2 y e -i t h H h ρ ⊗ |Ω Ω| e i t h H h .
Proof. In the stochastic presentation we can express the integral in ω in the definition of G h t as a partial trace

G h t (ρ) = ˆe-i t h H h,ω ρ1(ω)1(ω)e i t h H h,ω dP(ω) . = Tr L 2 (Ω P ,P) ˆ⊕ e -i t h H h,ω dP(ω)ρ ⊗ |1 1| ˆ⊕ e i t h H h,ω ′ dP(ω ′ ) .
Using the isomorphism

U := Id L 2 x ⊗ D : L 2 x ⊗ ΓL 2 y → L 2 x ⊗ L 2 (Ω P , P) we get U * ˆ⊕ e -i ∆t h H h,ω dP(ω) U = e -i ∆t h H h , and U * ρ ⊗ |1 1| U = ρ ⊗ |Ω Ω| with H h := U * ( ´⊕ H h,ω dP(ω)) U = -∆ x + √ 2hΦ 1 (τ x V ).
Hence the result.

4.3.

Existence of the dynamic. We show that the dynamic of the system is well defined. Since we work with a fixed h > 0 the value of h is here irrelevant and we set h = 1 in this section to clarify our exposition. We write for short • -∆ x for the operator -∆ x ⊗ Id ΓL 2 y , • N for the operator Id L 2

x ⊗ N with N = dΓ 1 (Id L 2 y ) the number operator on ΓL 2 y and

• Φ 1 (τ • V ) the operator on L 2 (R d ; ΓL 2 y ) ≃ L 2 (R d ) ⊗ ΓL 2 y defined by u → Φ 1 (τ • V )u with [Φ 1 (τ • V )u](x) := [Φ 1 (τ x V )][u(x)]. Proposition 4.4. If V belongs to the Sobolev space H 2 (R d ), then H = -∆ x + √ 2Φ 1 (τ • V ) , is essentially self-adjoint on D ′ := C ∞ 0 (R d ) ⊗ alg Γ F L 2
y and its closure is essentially self-adjoint on any other core for

N ′ = Id -∆ x + N .
Proof. We still denote by N ′ the closure of the essentially self-adjoint operator N ′ defined on D ′ . Then D ′ is a core for this operator. We remark that N ′ ≥ I on D ′ and thus also on D (N ′ ) as D ′ is a core for N ′ .

We verify the two estimates needed for Nelson's commutator theorem (see the book of Reed and Simon [START_REF] Reed | Methods of modern mathematical physics. II. Fourier analysis, self-adjointness[END_REF]). Let u ∈ D ′ , then

Hu L 2 x ⊗ΓL 2 y ≤ -∆ x u L 2 x ⊗ΓL 2 y + 2 V L 2 √ N + 1u L 2 x ⊗ΓL 2 y , ≤ (1 + 2 V L 2 ) N ′ u L 2 x ⊗ΓL 2 y .
In the sense of quadratic forms

[H, N ′ ] = √ 2[Φ(τ • V ), -∆ x + N ] = √ 2Φ(τ • ∇V ).∇ x + √ 2Φ(τ • ∆V ) + (a * (τ • V ) -a(τ • V )) so that | Hu, N ′ u -N ′ u, Hu | ≤ 6 V H 2 N ′1/
B] = ABA -1 . Let (D ε f ) (y) = ε -d/2 f y ε and H h,ε = Ad Id L 2 x ⊗ ΓD ε [H h ] = -∆ x ⊗ I Γy + √ 2hΦ ε -d/2 V (x -y ε ) . 5.
An approximated equation and its solution 5.1. Space translation in the fields and Fourier transform.

Notation 5.1. For an object X = (X 1 , . . . , X d ) with d components, like ξ ∈ R d , D x = (∂ x1 , . . . , ∂ x d ) or dΓ ε (D y ), let X .2 := X 2 1 + • • • + X 2 d .
We want to work with a field operator with no dependence in x. Then we recall that the translation τ x of x can be written as e -ix.Dy and thus

Γ e iεx.Dy H h,ε Γ e -iεx.Dy = (dΓ ε (D y ) -D x ) .2 + √ 2Φ ε ε -d/2 h ε V -y ε
where we use the ε-dependent operator dΓ ε . A conjugation by the Fourier transform in both the particle and the field variables yields a new expression for the Hamiltonian, and an approximated version

Ĥh,ε = ξ .2 -dΓ ε (2ξ.η) + dΓ ε (η) .2 + √ 2Φ ε (f h,ε ) , Ĥapp h,ε = ξ .2 + dΓ ε εη .2 -2ξ.η + √ 2Φ ε (f h,ε ) , with f h,ε (η) = ε d/2 h ε V (-εη), i.e. Ĥh,ε = Q W ick h,ε and Ĥapp h,ε = Q app,W ick h,ε with Q h,ε (z) = ξ .2 + z, (εη .2 -2ξ.η)z + z, ηz .2 + 2ℜ z, f h,ε , Q app h,ε (z) = ξ .2 + z, (εη .2 -2ξ.η)z + 2ℜ z, f h,ε .
Note that in the approximated Hamiltonian we neglect the quartic part z, ηz .2 . The evolution associated with the approximated Hamiltonian is explicitely solvable.

Definition 5.2. For ρ ∈ L 1 L 2 x , let ρ t = Ad e -i t ε H h,ε [ρ ⊗ |Ω Ω|] , ρ app t = Ad e -i t ε H app h,ε [ρ ⊗ |Ω Ω|] , ρt = Ad e -i t ε Ĥh,ε [ρ ⊗ |Ω Ω|] , ρapp t = Ad e -i t ε Ĥapp h,ε [ρ ⊗ |Ω Ω|] , ρ ε t = Tr ΓL 2 x [ρ t ] , ρ ε,app t = Tr ΓL 2 x [ρ app t ]
. This definition is consistent with the previous one given for ρ h t as ρ h t = ρ ε ε h t and the dilatation acts only in the Fock space part of L 2

x ⊗ ΓL 2 y . 5.2. The solution of the approximated equation.

Proposition 5.3. For ψ 0 ∈ L 2 x , let Ψh,ε,t = e -i t ε Ĥh,ε Ω ⊗ ψ0 and Ψapp h,ε,t = e -i t ε Ĥapp h,ε Ω ⊗ ψ0 , (5.1) z h,ε,t = -i ˆt 0 e -i s ε (ε 2 η .2 -2ξ.εη) f h,ε ds (5.2) ω h,ε,t = tξ 2 + ˆt 0 ℜ z s , f h,ε ds . (5.3) Then (1) Ψapp h,ε,t = e -i ω h,ε,t ε W √ 2 iε z h,ε,t Ω ⊗ ψ0 . (2)
There is a constant C G,d depending on G and the dimension d such that

|η| ν z h,ε,t L 2 η ≤ C G,d ( ht ε ) 1/2 ε 1/2-ν .
(

) Let T 0 > 0. There is a constant C T0,G,d such that for ht ε ≤ T 0 , Ψh,ε,t -Ψapp h,ε,t ≤ C T0,G,d ht ε / √ h 2 . 3
(4) For both Ψ♯ h,ε,t = Ψh,ε,t and Ψ♯ h,ε,t = Ψapp h,ε,t

(ε + N ε ) 1/2 Ψ♯ h,ε,t ≤ C d √ ε + t 2 ht ε Ĝ L 1 .
First we get rid of the quadratic part dΓ ε . Let

• Ψh,ε,t = e i t ε ξ .2 e i t ε dΓε(εη .2 -2ξ.η) Ψh,ε,t and Ψapp h,ε,t = e i t ε ξ .2 e i t ε dΓε(εη .2 -2ξ.η) Ψapp h,ε,t , • fh,ε,t = e i t ε (ε 2 η .2 -2ξ.εη) f h,ε , • zh,ε,t = -i ´t 0 fh,ε,s ds, • ωh,ε,t = ´t 0 ℜ zh,ε,s , fh,ε,s ds.
It is then enough to prove the results with the objects with a ∼ sign.

Lemma 5.4. Then Ψt (resp. Ψapp t ) is solution of the equation

iε∂ t Ψh,ε,t = QW ick h,ε Ψh,ε,t (resp. iε∂ t Ψapp h,ε,t = Qapp,W ick h,ε Ψapp h,ε,t )
with initial condition Ω ⊗ ψ0 , Qh,ε,t (z) = 2ℜ z, fh,ε,t + z, ηz .2 (resp. Qapp h,ε,t (z) = 2ℜ z, fh,ε,t ).

The function zh,ε,t is the solution of i∂ t zh,ε,t = ∂ z Qapp h,ε,t (z h,ε,t ) = fh,ε,t , with initial condition zh,ε,0 = 0 Proof. Indeed

iε∂ t Ψt = iε∂ t [e i t ε ξ .2 e i t ε dΓε(εη .2 -2ξ.η) Ψt ] = e i t ε ξ .2 e i t ε dΓε(εη .2 -2ξ.η) [2ℜ z, f + z, ηz .2 ] W ick Ψt = [2ℜ z, e it(εη .2 -2ξ.η) f + z, ηz .2 ] W ick e i t ε ξ .2 e i t ε dΓε(εη .2 -2ξ.η) Ψt = QW ick t Ψt .
And we can proceed analogously with Ψapp t .

Proof of Proposition 5.3. Point (1) follows from applying iε∂ t to the right hand side:

iε∂ t e -i ωt ε W √ 2 iε zt Ω ⊗ ψ0 = ∂ t ω -iε iε 2 ℑ √ 2 iε zt , - √ 2 ε ft + iεiΦ ε - √ 2 ε ft e -i ω t ε W √ 2 iε zt Ω ⊗ ψ0 = ∂ t ω -ℑ 1 i zt , ft + √ 2Φ ε ( ft ) Ψapp t since 1 t ϕ, [W (z + tu) -W (z)]ψ ---→ t→0 ϕ, [-iε 2 ℑ z, u + iΦ ε (u)]W (z)ψ .
For Point (2) we compute

|η| ν zh,ε,t 2 L 2 η = ˆt 0 ˆt 0 ˆRd η e i s-s ′ ε (ε 2 η .2 -2ξ.εη) |η| 2ν |f h,ε (η)| 2 dη ds ds ′ .
Note that the internal integral is uniformly bounded by

C G ε -2ν h ε . The change of variable η ′ = εη -ξ gives ˆRd η e i s-s ′ ε (ε 2 η .2 -2ξ.εη) |η| 2ν |f h,ε (η)| 2 dη = ε -2ν h ε e -i s-s ′ ε ξ .2 ˆRd η e i s-s ′ ε η .2 |η + ξ| 2ν Ĝ(η + ξ) dη as f h,ε (η) = ε d/2 h ε V (-εη) and h ε Ĝ(εη) ε d = |f h,ε (η)| 2 . For s = s ′ ˆRd η e i s-s ′ ε η .2 |η + ξ| 2ν Ĝ(η + ξ) dη = πε s ′ -s d/2 F (η → |η + ξ| 2ν Ĝ(η + ξ)) L 1 = πε s ′ -s d/2 F (η → |η| 2ν Ĝ(η)) L 1
The bound

|η| ν zh,ε,t 2 L 2 η ≤ C G h ε ε -2ν ˆt 0 ˆt 0 min πε s ′ -s d/2 , 1 ds ds ′ ≤ C G h ε ε -2ν π d/2 ε d/2 ˆ|s-s ′ |≥2δ,s,s ′ ∈[0,t] ds ds ′ (s ′ -s) d/2 + 2 √ 2tδ ≤ C G h ε ε -2ν π d/2 ε d/2 2 d/4 2 √ 2t 2 d-2 δ 1-d/2 + 2 √ 2tδ
is optimal when δ = ε.

For Point (3), let ∆ Qt (z) = z, ηz .2 . First we remark that

∆ Ψh,ε,t = - i ε ˆt 0 e -i t-s ε QW ick h,ε ∆ QW ick Ψapp h,ε,s ds . Since iε∂ t ∆ Ψt = QW ick ∆ Ψt + ∆ QW ick Ψapp t
and that the integral expression on the right satisfies the same differential equation. The difference ∆ Ψh,ε,t can then be controlled as

∆ Ψh,ε,t ≤ 1 ε ˆt 0 ∆ QW ick E(z h,ε,s ) ds .
The relation

E(z), R W ick E(z) = R(z) with R W ick = (∆ QW ick ) * ∆ QW ick gives Symb W ick ( z, ηz .2 ) W ick 2 = ( z, ηz .2 ) 2 + 4ε( z, ηz . ηz|)(|ηz . z, ηz ) + 2ε 2 ( ηz| .⊗2 )(|ηz .⊗2 ) ,
using the estimate in Point (2), we obtain that

∆ QW ick E(z h,ε,t ) 2 ≤ C T0,G,d ( ht ε ) 4 + 4ε ht ε 2 ht ε 2 + 2ε 2 ( ht ε 2 ) 2 which gives the result for ht ε ≤ T 0 . For Point (4), let γ t = (ε + N ε ) 1/2 Ψ♯ t , then iε∂ t (γ 2 t ) = Ψ♯ t , [Φ ε (f h,ε ), N ε ] Ψ♯ t with f h,ε = h ε ε d/2 V (εη), since ξ and dΓ ε (η) commute with N ε = dΓ ε (Id). We get [a ε (f h,ε ), dΓ ε (1)] = i∂ s Γ(e iεs ) a ε (f h,ε ) Γ(e -iεs ) s=0 = a ε (εf h,ε ) .
The other term of the commutator can be computed analogously (but

a ε (•) is C- antilinear whereas a * ε (•) is C-linear).
Introducing this relation into the differential equation and taking the modulus, we get

iε∂ t (γ 2 t ) ≤ √ 2 -1 Ψ♯ t a ε (εf h,ε ) Ψ♯ t + a * ε (εf h,ε ) Ψ♯ t . But a * ε (εf h,ε ) Ψ♯ t 2 ≤ εf h,ε 2 L 2 ξ Ψ♯ t , (ε + N ε ) Ψ♯ t
and the same estimate holds for annihilation operators.

Using Ĝ L 1 = h ε f h,ε 2 L 2 ξ
, we finally get a differential inequality for the function γ t

2εγ t ∂ t γ t ≤ iε∂ t γ 2 t ≤ 2εh Ĝ L 1 γ t .
The result follows by dividing by 2εγ t and integrating in time, since γ 0 = C d √ ε.

6. Measure of an observable at a mesoscopic scale for the approximated dynamics 6.1. Result. In this section we make the connection between the microscopic dynamic and the linear Boltzmann equation. α) for some constant C b,µ > 0 and µ (d, α) > 0. Remark 6.2. This result also holds with b a symbol in C ∞ 0 (R d * ξ ; C). The proof is the same as for Proposition 6.1, with the symplectic Fourier transform F σ replaced by the usual Fourier transform. The special case when b (ξ) = b 1 (|ξ| 2 ) is of particular interest and the symbol b t in the previous statement does not depend on t. Proposition 6.1 is a by-product of the following stronger result.

Proposition 6.1. Let α ∈ [0, 1) and assume h α ≤ ht ε ≤ 1. Let b ∈ C ∞ 0 (R d x × R d * ξ ) and ρ ∈ L + 1 L 2 x ,
′ ) = 2π Ĝ(ξ ′ -ξ) = 2π| V (ξ -ξ ′ )| 2 . The inequality m h (b, ρ ε,app t ) ≥ m h b ht ε , ρ -E 6 then holds with E 6 = C b,µ ht ε ht ε + h + h( ht ε ) -1 d/2-1 + h µ(d,
Proposition 6.3. Let b s ∈ C 1 (R; C ∞ 0 (R 2d x,ξ
)) such that for some R > 1, and for all s,

Supp ξ b s ⊂ B R \ B R -1 . Let ρ ∈ L + 1 L 2 x , Tr ρ ≤ 1 such that the kernel of ρ = Ad{F x }[ρ] has a bounded support. Then m h b ht ε , ρ ε,app t ≥ m h (b, ρ) -i ε ˆt 0 m h iε∂ s b s -ih{b s , ξ .2 } + ihQ -ht ε b s , ρ ε,app s ds -E 6 .
Remark 6.4. The conservation of the support in ξ is important and is provided by the properties of the dual linear Boltzmann equation in the application of this proposition.

Proof that Proposition 6.3 implies Proposition 6.1. Since one can make mistakes between the notations of those two propositions we use notations with tildes, b for Proposition 6.1 and without tildes for Proposition 6. Denote by G(t, t 0 ) the dynamical system associated with (-2ξ.∂ x -Q -t ) t given by

∂ t b t = (-2ξ.∂ x -Q -t ) b t b t=t0 = b 0 , b t = G(t, t 0 ) b 0 .
To have a vanishing term for b in the integral we require b ht/ε = G( ht ε , 0)b, so that with bht/ε = G(0, -ht ε ) b, we will get the expected result. The only thing remaining to prove is G(0, -t) = e tQ e 2tξ.∂x . It is equivalent to show that e 2tξ.∂x G (t, 0) = e -tQ , which is clear by derivation and using that Q t = e t2ξ.∂x Qe -t2ξ.∂x . 6.2. Expression of the measure of an observable for the approximated equation. We carry out an explicit computation using only the approximated equation.

Notation 6.5. Let σ(X 1 , X 2 ) = ξ 1 .x 2 -x 1 .ξ 2 (X j = (x j , ξ j ) ∈ R 2d x,ξ ) be the standard symplectic form on R 2d x,ξ . Let X ′ = (x ′ , ξ ′ ) ∈ R 2d
x,ξ , the Weyl operators on L 2 x are defined by

τ h X ′ = e -iσ(•,X ′ ) W (hx, D x ) = e -iσ(•,X ′ ) W (hx,Dx) = e i(ξ ′ •hx-x ′ •Dx) ,
their Fourier transform is denoted by τ h P := Ad {F x } τ h P . Note that the formula

τ h X1 τ h X2 = e i 2 hσ(X1,X2) τ h X1+X2 = e ihσ(X1,X2) τ h X2 τ h X1 holds. The symplectic Fourier transform F σ on L 2 (R 2d x,ξ ; C) is, with dX = dX/(2π) d , F σ b(X) = ˆR2d e -iσ(X,X ′ ) b(X ′ ) dX ′ . Note that (F σ ) -1 = F σ . Proposition 6.6. Let b be a symbol in C ∞ 0 (R 2d x,ξ ) and ρ ∈ L + 1 L 2 x , Tr ρ ≤ 1, then m h (b, ρ ε,app t ) = ˚F σ b(P ) K P (ξ 1 , ξ 2 ) τ h P (ξ 2 , ξ 1 ) dξ 1 dξ 2 dP
where

K P (ξ 1 , ξ 2 ) = e -i ω ξ 1 h,ε,t -ω ξ 2 h,ε,t ε ρ(ξ 1 , ξ 2 )e -1 2ε |z ξ 1 h,ε,t | 2 +|z ξ 2 h,ε,t | 2 +2 z ξ 2 h,ε,t ,e ipx •εη z ξ 1 h,ε,t . Let b t ∈ C 1 (R; C ∞ 0 (R 2d x,ξ )), then iε∂ t m h (b t , ρ ε,app t ) = m h (iε∂ t b t , ρ ε,app t ) + ih m {,} -m -+ m + .
where, for κ = {, } , -, + we define

m κ = ˆR2d P F σ b(P ) Tr ρapp t Γ(e ipx•εη ) A κ,P dP
with the operators A κ,P defined by their kernels, for j = 1, 2, by

A {,},P = A 1 {,},P -A 2 {,},P , ih A j {,},P (ξ 1 , ξ 2 ) = τ h P (ξ 2 , ξ 1 ) ∂ t ω ξj t , (6.1) A -,P = A 1 -,P + A 2 -,P , ih A j -,P (ξ 1 , ξ 2 ) = τ h P (ξ 2 , ξ 1 ) i∂ t 1 2 |z ξj t | 2 , (6.2) ih A +,P (ξ 1 , ξ 2 ) = τ h P (ξ 2 , ξ 1 ) i∂ t [ϕ, p x ] 2 , (6.3) with [ϕ, p x ] 2 = z ξ2
t , e ipx•εη z ξ1 t .

The indexes {, },and + are chosen to recall the terms of the linear Boltzmann equation, {, } corresponding to {ξ 2 , •}, + to Q + andto Q -. Remark 6.7. Each of those terms m κ is shown in the sequel to be of the form m κ = m(c κ , ρ ε,app t ) + ∆ κ where ∆ κ denotes a "small" error term.

Proof. Since b W (hx, D x ) = ´F σ b(P ) τ h P dP , we have for ρ ∈ L + 1 m h (b, ρ) = ˆF σ b(P ) Tr τ h P ρ dP .
From e iεx.λ τ h P e -iεx.λ = e iελ.px τ h P and taking λ as the spectral parameter of dΓ ε (D y ), Γ(e iεx.Dy ) τ h P Γ(e -iεx.Dy ) = Γ(e ipx•εDy ) τ h P and after conjugating with the Fourier transforms, we obtain

Ad (F x ⊗ ΓF y ) Γ(e iεxDy ) τ h P = Γ(e ipx•εη ) τ h P .
Thus, by translating and Fourier transforming we get the expression m h (b, ρ ε,app t ) = ˆF σ b(P ) Tr ρapp t Γ(e ipx•εη ) τ h P dP .

It then remains to compute the kernel K P of the operator

Tr ΓL 2 η [ρ app t Γ(e ipx•εη )] on L 2 ξ . Using ρ ⊗ |Ω Ω| = ´⊕ ξ1 ´⊕ ξ2 ρ(ξ 1 , ξ 2 ) |Ω Ω| dξ 1 dξ 2 we get Tr ΓL 2 η ρapp t Γ(e ipx•εη ) = Tr ΓL 2 η ˆ⊕ R d ξ 1 ˆ⊕ R d ξ 2 E(z ξ1 t ) E(z ξ2 t ) e -i ω ξ 1 t ε e i ω ξ 2 t ε ρ(ξ 1 , ξ 2 ) dξ 1 dξ 2 Γ(e ipx•εη )
and we obtain the kernel

K P (ξ 1 , ξ 2 ) = e -i ω ξ 1 t -ω ξ 2 t ε ρ(ξ 1 , ξ 2 ) E(z ξ2 t ) Γ(e ipx•εη ) E(z ξ1 t )
which brings the expected expression using the calculus on coherent states. For the formula for the derivative

iε∂ t m h (b, ρ ε,app t ) = ˚ F σ iε∂ t b(P ) + F σ b(P ) ∂ t ω ξ1 t -ω ξ2 t -i 1 2 ∂ t |z ξ1 t | 2 + |z ξ2 t | 2 + i∂ t [ϕ, p x ] 2 K P (ξ 1 , ξ 2 )τ h P (ξ 2 , ξ 1 ) dξ 1 dξ 2 dP
and so it suffices to observe that for κ = {, }, -, +,

Tr ρapp t Γ(e ipx•εη ) A κ,P = ¨ρ(ξ 1 , ξ 2 ) E(z ξ2 t ) Γ(e ipx•εη ) E(z ξ1 t ) e -i ω ξ 1 t -ω ξ 2 t ε A κ,P (ξ 1 , ξ 2 ) dξ 1 dξ 2 = ¨Aκ,P (ξ 1 , ξ 2 ) K P (ξ 1 , ξ 2 ) dξ 1 dξ 2 .
which is the expected result. 6.3. Two estimates. We need estimates to get rid of the term Γ(e ipx•εη ) and control errors on the operators A P . Proposition 6.8. Let A P be a P -dependent family of operators in L(L 2 ξ ). Then there exists a constant C G,d such that

P -k Tr ρapp t Γ(e ipx•εη ) -Id A P ≤ C G,d ht ε sup P ∈R 2d P -k A P L(L 2 ξ ) and ˆR2d P F σ b(P ) Tr ρapp t Γ(e ipx•εη ) -Id A P dP ≤ C G,d ht ε • k F σ b L 1 P sup P P -k A P L(L 2 ξ
) . This can be proved in two steps. Remark 6.9. It suffices to prove this property with ρ = |ψ ψ| with a ψ with bounded support as any ρ ∈ L + 1 L 2 x , Tr ρ = 1 the decomposition ρ = j≥0 λ j |ψ j ψ j | holds with positive λ j 's and j λ j = 1, and

Supp ρ(ξ, ξ ′ ) ⊂ B 2 M ⇔ ∀j, Supp ψj ⊂ B M .
Proof. For Ψ be a normed vector in

L 2 ξ ⊗ ΓL 2 η Tr | Ψ Ψ| Γ(e ipx•εη ) -Id A P ≤ Γ(e ipx•εη ) -Id Ψ A P L(L 2 ξ ) . For Ψ = Ψapp
h,ε,t associated with ψ, the calculus on coherent states gives

Γ(e ipx•εη ) -Id Ψapp h,ε,t 2 = sup ξ E(e ipx•εη z ξ h,ε,t ) -E(z ξ h,ε,t ) 2 = sup ξ 2 1 -cos 1 ε ℑ e ipx.εη z ξ h,ε,t , z ξ h,ε,t ≤ C 2 G,d ( ht ε ) 2 ,
where the inequality follows form |1cos t| ≤ t 2 /2 and the estimates on z t . We then get the second result by an integration. Proposition 6.10. Let E P be a P -dependent family of operators in L(L 2 ξ ) and ρ be a state on L 2 ξ ⊗ ΓL 2 η . Then for any integer k (with possibly infinite quantities) ˆR2d

P F σ b(P ) |Tr [ρ E P ]| dP ≤ • k F σ b L 1 P sup P P -k E P L(L 2 ξ ) .
6.4. The transport term m {,} . The result of this section is the following. Proposition 6.11.

Let ρ ∈ L + 1 L 2 x , Tr ρ ≤ 1 and b ∈ C ∞ 0 (R 2d x,ξ ) such that Supp ρ(ξ, ξ ′ ) ⊂ B 2 R , and Supp ξ b ⊂ B R for some R > 0 then m {,} = m(-{b, ξ .2 }, t) + ∆ {,} with |∆ {,} | ≤ C G,R,b ( ht ε + h + ( ε t ) d/2
). Remark 6.12. We can introduce a cutoff function

χ R ∈ C ∞ 0 (R d ξ ) such that χ R (B R ) = {1}, χ R (R d ξ \ B R+1 ) = {0} and χ R (R d ξ ) ⊂ [0, 1]. Proposition 6.
11 is proved by doing a succession of approximations. The error terms ∆ {,},j , j = 1, 2, 3 are given by the approximation process (where we write shortly b W for b W (-hD ξ , ξ))

m {,} = ˆF σ b(P ) Tr ρapp t Γ(e ipx•εη ) 1 ih τ h P , χ R ∂ t ω× dP = Tr ρapp t 1 ih b W , χ R ∂ t ω× dP + ∆ {,},1 = ˆF σ -{b , ξ .2 } (P ) Tr ρapp t τ h P dP + 2 j=1 ∆ {,},j = m -{b , ξ .2 }, t + 3 j=1 ∆ {,},j .
where we used that A {,},P = 1 ih [τ h P , ∂ t ω×] and where the quantities ∆ {,},j are defined by

∆ {,},1 = ˆF σ b(P ) Tr ρapp t Γ(e ipx•εη ) -Id 1 ih τ h P , χ R ∂ t ω× dP , ∆ {,},2 = Tr ρapp t 1 ih b , χ R ∂ t ω × -h i {b , χ R ξ .2 } W dP , ∆ {,},3 = ˆF σ -{b , ξ
|∆ {,},1 | ≤ 2 ht ε • k F σ b L 1 P O 1 + h + [h( ht ε ) -1 ] d/2-1 , (2) |∆ {,},2 | ≤ F σ b L 1 P + • k F σ b L 1 P O h + ( ε t ) d 2 -1 , (3) |∆ {,},3 | ≤ ht ε F σ {b, ξ .2 } L 1 P .
Proof of Proposition 6.13. Point 1 is a result of Proposition 6.8 and Lemma 6.14.

For Point 2

∆ {,},2 = ˆR2d P F σ b(P ) Tr ρapp t 1 ih τ h P , χ R ∂ t ω h,ε,t × - h i τ h P , χ R ξ .2 W dP
so that Lemma 6.14 and Proposition 6.10 give the estimation. Point 3 is an application of Proposition 6.8.

Lemma 6.14. We have, for some integer k,

[τ h P ,χ R ∂ t ω h,ε,t ×] = -ih{e iσ(P,X) , χ R ξ 2 } W (-hD ξ , ξ) + hO P k h + ( ε t ) d 2 -1 .

and in particular τ

h P , χ R ∂ t ω× L(L 2 ξ ) ≤ P k O(h).
Proof of Lemma 6.14. First observe that the time derivative of ω is given by

∂ t ω h,ε,t = ξ .2 + ℜ z ξ h,ε,t , f h,ε = ξ .2 -hℑ ˆt/ε 0 ˆRd η e is(η .2 -2ξ.η) Ĝ(η) dη ds
once we replace f h,ε by its expression in terms of V , use Ĝ = | V | 2 and make a change of variable. By setting

R(u, ξ) := χ R (ξ)ℑ lim M→+∞ ˆM u ˆRd η e is(η .2 -2ξ.η) Ĝ(η) dη ds we get χ R ∂ t ω = χ R (ξ)ξ .2 -hR(0, ξ) + hR( t ε , ξ).
The part in ξ .2 gives the only relevant contribution

[τ h P , χ R ξ .2 ×] = -ih{e iσ(P,X) , χ R ξ .2 ×} W eyl + P k O h→0 (h 2
) . One of the other parts can be estimated without using the commutator structure

[τ h P , R( t ε , ξ)×] L(L 2 ξ ) ≤ 2 τ h P L(L 2 ξ ) R( t ε , ξ) L ∞ ξ ≤ C( ε t ) d 2 -1 since ˆRd η e is(η .2 -2ξ.η) Ĝ(η) dη = e -isξ .2 ˆRd x G(x) e -ix.ξ 2π |s| d/2 e id sign s π 4 e x 2
2is dx whose modulus is bounded by

( 2π |s| ) d/2 G L 1 . Since R(0, •) is in C ∞ 0 (R d ξ )
we can apply the symbolic calculus [τ h P , hR(0, ξ)×] = -ih 2 {e iσ(P,X) , R(0, ξ)} W (-hD ξ , ξ) + O(h 2 P k ) where for some integer k, {e iσ(P,X) , R(0, ξ)×} W (-hD ξ , ξ) L(L 2 ξ ) = P k O h→0 (1) , which concludes the proof of the lemma.

The collision terms m

-and m + . Proposition 6.15. Let b ∈ C ∞ 0 (R d x × R d * ξ ) and ρ ∈ L + 1 L 2 x , Tr ρ ≤ 1 such that for some R > 0, Supp ξ b ⊂ B R -B 1/R and Supp ρ(ξ, ξ ′ ) ⊂ B 2 R . Then m ± = m(Q ±,t (b), t) + ∆ ± and for any α ∈ [0, 1), there are constants µ = µ(d, α) > 0 and C R,b,G,d,α,µ > 0, such that for h α ≤ th ε ≤ 1, |∆ ± | ≤ C R,b,G,µ ht ε + h µ . Notation For ζ > 0, r ∈ R and P ∈ R 2d px,p ξ , set, with κ ζ (r) = 1 π ζ r 2 +ζ 2 , c(ξ) = 2π ˆRd η Ĝ(η + ξ) δ(η .2 -ξ .2 ) dη , c ζ (ξ) = 2π ˆRd η Ĝ(η) κ ζ (η .2 -2ξ.η) dη , c ζ P,t (x, ξ) = 2π ˆRd η Ĝ(η) e iσ(P,(-2tη,-η)) κ ζ (η .2 -2ξ.η) dη .
Associate with these functions the operators defined for b

∈ C ∞ 0 (R d x × R d * ξ ) by Q ζ -(b) = c ζ b , Q -(b) = c b , Q ζ +,t b(x, ξ) = ˆR2d P
F σ b(P ) e iσ(P,X) c ζ P,t (x, ξ) dP . Proposition 6.16. For d ≥ 3, and

h α ≤ ht ε ≤ 1, m ± = m(Q ±,t (b), t) + 4 k=1 ∆ ±,k with • |∆ ±,1 | ≤ ht ε C d max Ĝ L 1 , G L 1 F σ b L 1 P , • |∆ ±,2 | ≤ C α,β,ν,G,d h ν , • |∆ ±,3 | ≤ ζ γ N k(d) (b) C d,G,C,γ for γ ∈ (0, 1), • |∆ ±,4 | ≤ ht ε F σ Q ±, ht ε (b) L 1 P for some ν, β > 0 with ζ = h β .
This result will be proved in the next paragraphs by considering successively all the error terms. These error terms ∆ ±,j , j = 1, . . . , 4 are given by the following approximation process (where we write shortly

B W for B W (-hD ξ , ξ)) m ± = ˆF σ b(P ) Tr ρapp t A ±,P dP + ∆ ±,1 = ˆF σ b(P ) Tr ρapp t c ζ ±,P e iσ(P,•) W dP + 2 j=1 ∆ ±,j = ˆF σ Q ±, ht ε b (P ) Tr ρapp t τ h P dP + 3 j=1 ∆ ±,j = m(Q ±, ht ε b, t) + 4 j=1 ∆ ±,j .
The error terms ∆ ±,j are thus given by ∆ ±,1 = ˆF σ b (P ) Tr ρapp t Γ(e ipx•εη ) -Id A ±,P dP , (6.4)

∆ ±,2 = ˆF σ b (P ) Tr ρapp t A ±,P -c ζ
±,P e iσ(P,•) W dP , (6.5)

∆ ±,3 = Tr ρapp t Q ζ ±, ht ε b -Q ±, ht ε b W , (6.6) ∆ ±,4 = ˆF σ Q ±, ht ε b (P ) Tr ρapp t Id -Γ(e ipx•εη ) τ h P dP , (6.7) since τ h P = e iσ(P,•) W , Q ζ ±, ht ε b W = ˆR2d P F σ b(P ) c ζ ±,P e iσ(P,•) W dP ,
and the same relation holds without ζ and

ˆP F σ Q ±, ht ε b (P ) Tr ρapp t Γ(e ipx•εη ) τ h P dP = m Q ±, ht ε b, t .
The term ∆ ±,4 can be estimated right away using Proposition 6.8.

6.5.1. Computation of the operators A ±,P . We recall that the operators A ±,P are defined by their kernels in Equations (6.1), (6.2), (6.3).

Proposition 6.17. The operators A -,j can be expressed as

A 1 -,P = ˆRd η ˆt/ε 0 τ h P • ℜ e is(η .2 -2ξ.η) × Ĝ(η) ds dη , A 2 -,P = ˆRd η ˆt/ε 0 ℜ e is(η .2 -2ξ.η) × • τ h P Ĝ(η) ds dη .
The operator A +,P can be decomposed as A +,P = A 1 +,P + A 2 +,P with

A 1 +,P = ˆt/ε 0 ˆRd η e -iσ(P,(2 t ε η,η)) τ h P • e -is(η .2 -2ξ.η) Ĝ(η) dη ds , A 2 +,P = ˆt/ε 0 ˆRd η e -iσ(P,(2 t ε η,η)) e is(η .2 -2ξ.η) • τ h P Ĝ(η) dη ds . Proof. Computing the time derivative of 1 2 |z h,ε,t | 2 brings ∂ t 1 2 |z h,ε,t | 2 = hℜ ˆRd η ˆt/ε 0 e is(η .2 -2ξj .η) Ĝ(η) ds dη .
From the definition of A -,j,P in terms of their kernel, we get

ih A 1 -,P = iτ h P • (∂ t 1 2 |z ξ t | 2 ) , ih A 2 -,P = i(∂ t 1 2 |z ξ t | 2 ) • τ h P , hence the result for A j -. The time derivative of [ϕ, p x ] 2 is ∂ t [ϕ, p x ] 2 = h ˆRd η ˆt/ε 0 e ipx•η e is(η .2 -2ξ1.η) e -i t ε (η .2 -2ξ2.η) ds Ĝ(η) dη + h ˆRd η ˆt/ε 0 e ipx•η e i t ε (η .2 -2ξ1.η) e -is(η .2 -2ξ2.η) ds Ĝ(η) dη .
We now focus on the first term (analogous computations give the second term). The definition of A +,P in terms of their kernel gives then

A 1 +,P = ˆt/ε 0 ˆRd η e ipx•η e -i t ε (η .2 -2ξ.η) • τ h P • e is(η .2 -2ξ.η) Ĝ(η) dη ds ,
The relation e 2i t ε ξ.η • τ h P = e -2i t ε p ξ η τ h P • e 2i t ε ξ.η brings the result up to a change of variable.

Thus we get six different terms (four for the A -terms due to the real parts and two for the A + terms) with a very similar structure. In order to avoid repeating analogous calculations several times we introduce the following notations.

We can then compute, for ψ, ϕ ∈ L 2 ξ ,

ψ, A j µ (s)ϕ = ˆRd η ψ, Ĝ(η) e iµ1 σ τ h P • e -µ2is(η .2 -2ξ.η) ϕ ξ dη = ˆRd ξ Ĝ(η) τ h -P ψ(ξ), e µ1iσ e -µ2is(η .2 -2ξ.η) ϕ(ξ) η dξ = ˆRd θ ψ θ , ϕ µ,θ ξ dθ/(2π) d ,
where we defined, for θ ∈ R d θ ,

ϕ µ,θ = ˆeiθη e µ1iσ e -µ2is(η .2 -2ξ.η) ϕ(ξ) dη , ψ θ = ˆeiθη Ĝ(η) τ h -P ψ(ξ) dη .
We first compute

ϕ µ,θ (ξ) = ( π s ) d/2 e i (θ+µ 2 2sξ+µ 1 (2hsp ξ -px )) 2 4µ 2 s e i π 4 d ϕ(ξ)
where we used the formula ´e-ixη e -aη 2 dη = π a d/2 e -x 2 /4a with a = µ 2 is and x =

-(θ + µ 2 2sξ + µ 1 (2hsp ξ -p x )) and so ϕ µ,θ L ∞ (R d θ ;L 2 ξ ) ≤ π s d/2 ϕ L 2 ξ . We now observe that ˆeiθη Ĝ(η) τ h -P dη L 1 (R d θ ;L(L 2 ξ )) ≤ (2π) d G L 1 so that ψ θ L 1 (R d θ ;L 2 ξ ) ≤ C d G L 1 ψ L 2 ξ
. And finally

ψ, A µ (s)ϕ ≤ C d G L 1 ( π s ) d/2 ϕ L 2 ξ ψ L 2 ξ
and we obtain the desired result

A µ (s) L(L 2 ξ ) ≤ C d G L 1 s -d/2 . 6.5.3. Estimate of the error terms ∆ ±,2 . Proposition 6.21. Let α ∈ (0, 1]. There are constants β = β(d, α) ∈ (0, 1), ν = ν(d, α) ∈ (0, 1) and C = C(α, β, ν, d, G) > 0 such that, for h α ≤ th ε ≤ 1, and ζ = h β , |∆ ±,2 | ≤ • k F σ b L 1 Ch ν .
In order to prove this result we use Proposition 6.10 and thus control

ˆt/ε 0 A (s) ds -C ζ L(L 2 ξ )
.

We first give an abstract result and then show that our cases fit within this framework.

Proposition 6.22. For M , t, ε such that 1 ≤ M ≤ t ε . Suppose given (A (s)) s≥0 , (B(s)) s≥0 and (C ζ ) 0<ζ<1 three families of operators in L(L 2 ξ ) (also dependent on h and P = (p x, p ξ )) such that for some constants

C A , C A ,B , C B,C , independent of h, ε, t, P, M, ζ, (1) A (s) L(L 2 ξ ) ≤ C A min 1, s -d/2 , (2) A (s) -B(s) L(L 2 ξ ) ≤ C A ,B hs |p ξ |, (3) r ζ,M (x, ξ) := Symb W eyl ( ´M 0 B(s) e -ζs ds -C ζ ) satisfies for some k = k(d) ∈ N, sup |α|≤k ∂ α x,ξ r ζ,M L ∞ x,ξ ≤ C B,C P k M ζ k e -ζM .
Then, for ζM ≥ 1,

/ε 0 A (s) ds L(L 2 ξ ) ≤ d d-2 C A , (2) ´t/ε 0 A (s) ds -´M 0 A (s) ds L(L 2 ξ ) ≤ 2 d-2 C A M 1-d 2 , (3) for d ≥ 3, ˆM 0 A (s) 1 -e -ζs ds L(L 2 ξ ) ≤ 5C A ζ 1/2 (4) ´M 0 (A (s) -B(s)) e -ζs ds L(L 2 ξ ) ≤ 1 2 C A ,B hζ -2 |p ξ |, (5) for some integer k = k(d), ˆM 0 B(s) e -ζs ds -C ζ L(L 2 ξ ) ≤ C d,k ′ C B,C P k M ζ k e -ζM . (6) Let ht ε ≥ h α , ζ = h β with β ∈ (0, 1 2 ) and β + α < 1, and ν = ν(α, β) < min{(1 -α)/2, β/2, 1 -2β}, we have ˆt ε 0 A (s) ds -C ζ L(L 2 ξ ) ≤ Ch ν with C = C(ν, α, β, C A , C A B , C BC ). (1) ´t
Proof. Points 1 and 2 are proved by integration of the first assumed estimate and using 1 ≤ M ≤ t ε for 2. Point 3 is proved by integration of the first assumed estimate, using 1-e -ζs ≤ ζs for ζs ≤ 1 and 1e -ζs ≤ 1 for ζs ≥ 1,

ˆM 0 (1 -e -ζs ) min{1, s -d/2 } ds ≤ ζ ˆ1 0 s ds + ζ ˆ1/ζ 1 s 1-d 2 ds + ˆ+∞ 1/ζ s -d/2 ds ,
which brings the result.

For Point 4, we use the second assumption and ´M 0 se -ζs ds ≤ ζ -2 ´+∞ 0 ue -u du . For Point 5, the known estimates for pseudo-differential operators give

r W (-hD ξ , ξ) ≤ C k sup |α|≤N k ∂ α x,ξ r L ∞ (R 2d ) .
This and the third hypothesis imply the result. For Point 6, we would like to choose the (h-dependent) parameters M and ζ such that the quantity

M 1-d 2 + ζ + hζ -2 + M ζ k e -ζM ,
is small when h tends to 0 and M not too big. We choose hM = h α and ζ = h β with β + α < 1, α, β > 0 so that the previous quantity is smaller than

h (1-α)( d 2 -1) + h β/2 + h 1-2β + h -k(1-α+β) exp -(h β+α-1 ) .
In order to get a small quantity it suffices to require β < 1 2 . Then we get an error term whose size is controlled by h ν(α,β) .

C A = C d max Ĝ L 1 , G L 1 , C A ,B = |•| Ĝ L 1 , C B,C = • k Ĝ L 1 , for some integer k.
Proof. Point 1 is contained in Proposition 6.20.

We show Point 2 for A 1 µ and B 1 µ , the proof can be adapted to the case of A 2 µ and B 2 µ . We observe that τ h P • e µ2is2ξ.η × = e -µ2isηhp ξ τ h P -(µ22sη,0) and e iσ(P,X) e µ2is2ξ.η W (-hD ξ , ξ) = τ h (px-µ22sη,p ξ ) . Thus we obtain the estimation Proof. We recall that

τ h P • e µ2is2ξ.η × -e iσ(P,X) e µ2is2ξ.η W (-hD ξ , ξ) L(L 2 ξ ) ≤ hs |η| |p ξ | Since the Weyl symbol of B 1 µ (s) is 1 2 ˆRd η Ĝ(η) e
∆ -,3 = Tr ρapp t (Q ζ -b -Qb) W (-hD ξ , ξ -dΓ ε (η)) so that |∆ -,3 | ≤ (Q ζ -b -Q -b) W (-hD ξ , ξ -dΓ ε (η)) L(L 2 ξ ⊗ΓL 2 η ) ≤ C k,d N k (Q ζ -b -Q -b)
for some integer k big enough.

∂ α ξ c ζ -c (ξ) ≤ C k,γ,G,R ζ γ .
Proof. With κ ζ , c, c ζ introduced in Definition 6.5, c ζc can be expressed as

c ζ -c (ξ) = ˆRd η Ĝ(η) κ ζ (η .2 -2ξ.η) dη - ˆRd η Ĝ(ξ + η) δ |η| 2 -|ξ| 2 dη .
We express the first integral as

ˆRd η Ĝ(η) κ ζ (η -ξ) .2 -ξ .2 dη = ˆSd-1 ˆRρ f ξ,ω (r) κ ζ (ξ .2 -r) dr dω = ˆSd-1 f ξ,ω * κ ζ (ξ .2 ) dω and f ξ,ω (r) := 1 2 r d-2 2 g(ξ + √ rω) 1 [0,+∞) (r).
The partial derivative

∂ ξj f ξ,ω (r) = 1 2 r d-2 2 ∂ ξj g(ξ + √ rω) 1 [0,+∞) (r)
has the same form as the function f ξ . Then we observe that

∂ ξj f ξ,ω * κ ζ -f ξ,ω |ξ| 2 = (∂ ξj f ξ,ω ) * κ ζ -∂ ξj f ξ,ω |ξ| 2 + ∂ r (f ξ,ω * κ ζ -f ξ,ω ) |ξ| 2 2ξ j
so that by doing successive derivations it suffices to deal only with quantities of the form

∂ k r (∂ β ξ f ξ,ω * κ ζ -∂ β ξ f ξ,ω ) which are in fact of the form ∂ k r (f * κ ζ -f
) with f satisfying the hypotheses of Lemma 6.26 uniformly in ω so that we get the expected control, by integration over ω.

Lemma 6.26. Let f : R r → R continuous, vanishing on R -, such that f | R + * ∈ C ∞ (R + * ) is rapidly decreasing towards +∞. Let 0 < r min < r max . Then ∀γ ∈ (0, 1) , ∃C f,γ , ∂ k r [f * κ ζ -f ] [rmin,rmax] L ∞ ≤ C γ ζ γ .
Proof. We choose A and ∆r such that 0 < A < ∆r < r min /2. Let χ 1 a C ∞ decreasing function such that Lemma 6.30 gives, for the second term,

χ 1 (r) = 1 if r ≤ A/2 = 0 if A ≤ r . Let f 1 = χ 1 f and f 2 = (1 -χ 1 ) f then f * δ ζ = f 1 * E ′ ,C ∞ κ ζ + f 2 * S,L 1 κ ζ . Since ∂ k r f 2 * κ ζ = ∂ k r f 2 * κ ζ ,
∂ k r f 2 * κ ζ -π∂ k r f 2 L ∞ ≤ C γ f (k) 2 ∞ + f (k+1) 2 ∞ ζ γ .
We are only interested in r ∈ [r min , r max ] with 0 < r min < r max when evaluating

∂ k r (f * κ ζ ). We insert another cutoff function χ 2 ∈ C ∞ 0 (R) such that χ 2 (r) = 0 if r ≤ r min -2∆r = 1 if r min -∆r ≤ r ≤ r max + ∆r = 0 if r max + 2∆r ≤ r Then f 1 * κ ζ = f 1 * χ 2 κ ζ + f 1 * (1 -χ 2 )
κ ζ and our hypotheses on the supports give

Supp{f 1 * (1 -χ 2 )κ ζ } ⊂ Supp f 1 + Supp(1 -χ 2 ) ⊂ R \ [r min -∆r + A, r max + ∆r] .
Since A < ∆r we obtain f 1 * (1χ 2 ) κ ζ [rmin,rmax] = 0 and we can restrict ourselves to the computation of

f 1 * E ′ ,C ∞ 0 χ 2 κ ζ .
More precisely we want to estimate

∂ k r f 1 * E ′ ,C ∞ 0 χ 2 u ζ [rmin,rmax] L ∞ since χ 2 δ = 0 and thus f 1 * E ′ ,E ′ χ 2 δ = 0.
But the same considerations hold for the supports of the derivatives. Thus it is sufficient to observe that we have the control

f 1 * L 1 ,C ∞ 0 ∂ k r (χ 2 κ ζ ) L ∞ ≤ f 1 L 1 ∂ k (χ 2 κ ζ ) L ∞ ≤ f 1 L 1 C χ2 sup r≥rmin-2∆r |∂ k κ ζ |
where the sup is controlled by Cζ with C only dependent on ∆r and r min since

2∂ k κ ζ (r) = i k k! -(ir -ζ) k+1 + (ir + ζ) k+1 (r 2 + ζ 2 ) k+1 . Consequently ∂ k r f 1 * E ′ ,C ∞ 0 χ 2 κ ζ -f 1 * E ′ ,E ′ χ 2 δ [rmin,rmax] L ∞ ≤ Cζ
and this ends the proof.

6.5.5. Estimate of the error term ∆ +,3 .

Remark 6.27. Throughout this section we will make definitions that are dependent on the value of th ε . This will not be a problem as long as th ε ≤ 1 which will be satisfied with our choice of ε = ε (h) ≫ h.

Proposition 6.28. Let b ∈ C ∞ 0 (R 2d x,ξ ) with Supp ξ b ⊂ B R \ B 1/R for some R > 1.
Let γ ∈ (0, 1). There exists a constant C G,R,γ > 0 such that, for all ζ > 0,

|∆ +,3 | ≤ ζ γ N k (b) C G,R,γ for some integer k = k(d) big enough. Proof. Since ∆ +,3 = Tr ρapp t Q ζ +, ht ε b -Q +, ht ε b W (-hD ξ , ξ -dΓ ε (η)) we get |∆ +,3 | ≤ Q ζ +, ht ε b -Q +, ht ε b W (-hD ξ , ξ -dΓ ε (η)) L(L 2 ξ ⊗ΓL 2 η ) ≤ C k,d N k Q ζ +, ht ε b -Q +, ht ε b for some integer k = k (d) big enough.
Thus we boil down to prove that for any integer k ≥ 0 there is a constant C k,b,G,γ > 0 such that for any ζ > 0

N k Q ζ +, ht ε b -Q +, ht ε b ≤ C k,G,γ N k (b) ζ γ .
But we have a convenient expression for

Q ζ +, ht ε Q ζ +, ht ε b(x, ξ) = 2π ˆRd η Ĝ(η) b x -2 ht ε η, ξ -η κ ζ (η .2 -2ξ.η) dη = 2π ˆRd η Ĝ(ξ -η) b x -2 ht ε ξ + 2 ht ε η, η κ ζ (η .2 -2ξ.η) dη = π ˆSd-1 ω ˆR+ r ϕ ω (x, ξ, r) K ζ (r -ξ .2
) dr dω , with ϕ ω (x, ξ, r) = 0 for r ≤ 0, and for r ≥ 0,

(6.14) ϕ ω (x, ξ, r) = Ĝ(ξ - √ rω) b x -2 ht ε ξ + 2 ht ε √ rω, √ rω r d/2-1
defined for ω ∈ S d-1 and x, ξ ∈ R d . We also have a convenient expression for

Q +, ht ε b in terms of ϕ ω , Q +, ht ε b (x, ξ) = π ˆSd-1 ω ϕ ω (x, ξ, ξ .2 ) dω .
The conclusion is then given by Lemma 6.29. Lemma 6.29. For any γ ∈ (0, 1), uniformly in ω ∈ S d-1 ω ,

N k ˆR+ r ϕ ω (x, ξ, r) κ ζ (r -ξ .2 ) dr -ϕ ω (x, ξ, ξ .2 ) ≤ C k,G,γ ζ γ .
Proof. The integral can be expressed as a convolution product ˆRr

ϕ ω (x, ξ, r) κ ζ (r -ξ .2 ) dr = ϕ(x, ξ, •) * κ ζ (ξ .2 ) .
Since the derivation behaves well with the difference, i.e.

∂ α x ∂ β ξ ϕ ω (x, ξ, •) * κ ζ (ξ .2 ) -ϕ ω (x, ξ, ξ .2 ) = α ′ ,β ′ ,γ ′ c α ′ ,β ′ ,γ ′ 2 |γ ′ | ξ γ ′ × ∂ α ′ x ∂ β ′ ξ ∂ γ ′ r ϕ ω (x, ξ, •) * κ ζ (ξ .2 ) -∂ α ′ x ∂ β ′ ξ ∂ γ ′ r ϕ ω (x, ξ, ξ .
2 ) , it suffices to apply Lemma 6.30.

∈ (0, 1), a constant C γ > 0 exists such that ∀ζ > 0, f * κ ζ -f L ∞ ≤ max { f ∞ , f ′ ∞ } C γ ζ γ .
Proof. The formula f (r 0 + ζr)f (r 0 ) = ζr ´1 0 f ′ (r 0 + sζr) ds and an interpolation with |f (r

0 + ζr) -f (r 0 )| ≤ 2 f ∞ give for γ ∈ [0, 1], |f (r 0 + ζr) -f (r 0 )| ≤ 2 max { f ∞ , f ′ ∞ } ζ γ |r| γ . So, for γ ∈ [0, 1), ˆR[f (r 0 + ζr) -f (r 0 )] dr r 2 +1 ≤ max{ f ∞ , f ′ ∞ }C γ ζ γ
which is the expected result.

7. Comparisons of the measures of an observable at a mesoscopic scale for the original and approximated dynamics

Remark 7.1. Let b ∈ C ∞ 0 (R 2d x,ξ ), ρ ∈ L 1 L 2 x and t ≥ 0, m(b, ρ ε t ) = Tr b W -hD ξ , ξ -dΓ ε (η) ρt , m(b, ρ ε,app t ) = Tr b W -hD ξ , ξ -dΓ ε (η) ρapp t . Definition 7.2. Let b ∈ C ∞ 0 (R 2d x,ξ ), ρ ∈ L 1 L 2 x a state, t ≥ 0 and χ ∈ C ∞ 0 (R 2d x,ξ ) we define m(b, ρ, t, χ) = Tr χ dΓ ε (η) b W -hD ξ , ξ -dΓ ε (η) χ dΓ ε (η) ρt m app (b, ρ, t, χ) = Tr χ dΓ ε (η) b W -hD ξ , ξ -dΓ ε (η) χ dΓ ε (η) ρapp t . Proposition 7.3. Assume ht ε / √ h. Let b ∈ C ∞ 0 (R 2d x,ξ ) non-negative such that Supp ξ b ⊂ B R \ B 1/R for some R > 0, ρ ∈ L + 1 L 2 x with Tr ρ ≤ 1 and for j = 1, 2, χ j ∈ C ∞ 0 (R d λ ) with values in [0, 1], χ j (B Mj ) = {1} for M 1 = 3R and with χ 2 (R d -B R+1 ) = {0}.
There is a constant C R,b,χ1,χ2 (which does not depend on ρ) such that

m app h b, (ρ χ2 ) app t -m h (b, ρ t ) ≤ E 7 = C R,b,χ1,χ2 h + ( ht ε / √ h) 3 + E 6 with ρ χ2 = χ 2 (D x ) ρ χ 2 (D x ).
We use the decomposition E 7 = E 7.1 + E 7.2 + E 7.3 corresponding to the steps:

(

1) m h (b, ρ χ2 , t, χ 1 ) -m h (b, ρ t ) ≤ E 7.1 = Ch, (2) m app h (b, ρ χ2 , t, χ 1 ) -m h (b, ρ χ2 , t, χ 1 ) ≤ E 7.2 = C( ht ε / √ h) 3 , (3) m h (b, (ρ χ2 ) app t ) -m app h (b, ρ χ2 , t, χ 1 ) ≤ E 7.3 = E 6 + Ch.

7.1.

Step 1: Introduction of cutoffs. We introduce cutoff functions both on the state ρ and the Wick observable b W (-hD ξ , ξ -dΓ ε (η)).

Proposition 7.4. Let b ∈ C ∞ 0 (R 2d x,ξ ) non-negative such that Supp ξ b ⊂ B R for some R > 0, ρ ∈ L + 1 L 2 x , Tr ρ ≤ 1, and, for j = 1, 2, χ j ∈ C ∞ 0 (R d λ ) with values in [0, 1] and χ j (B Mj ) = {1} for some M j > 0. Then there is a constant C b,χ1,χ2 such that m(b, ρ χ2 , t, χ 1 ) -m(b, ρ t ) ≤ E 7.1 = C b,χ1,χ2 h with ρ χ2 = χ 2 (D x ) • ρ • χ 2 (D x ).
Proof. Using the functional calculus for the self-adjoint operator dΓ ε (η) and since

b(x, ξ -λ) ≥ χ 2 (ξ) b(x, ξ -λ) χ 1 (λ) χ 2 (ξ) ≥ χ 2 (ξ) ♯ h b(x, ξ -λ) χ 1 (λ) ♯ h χ 2 (ξ) -C b,χ1,χ2 h holds uniformly in λ, we can write b W -hD ξ , ξ -dΓ ε (η) ≥ χ 2 (ξ) • b W -hD ξ , ξ -dΓ ε (η) χ 1 (dΓ ε (η)) • χ 2 (ξ) -C b,χ1,χ2 h . And thus m(b, ρ t ) = Tr b W -hD ξ , ξ -dΓ ε (η) ρt ≥ Tr b W -hD ξ , ξ -dΓ ε (η) χ 1 (dΓ(η)) ρ χ2 t -C b,χ1,χ2 h since [H ε , χ 2 ] = 0.
≤ √ h. Let b ∈ C ∞ 0 (R 2d x,ξ ) non-negative, ρ ∈ L + 1 L 2 x , Tr ρ ≤ 1 and χ ∈ C ∞ 0 (R d λ ) with values in [0, 1], and χ(B M ) = {1} for some M > 0, then there is a constant C G,b,χ such that |m(b, ρ, t, χ) -m app (b, ρ, t, χ)| ≤ E 7.2 = C G,b,χ ht ε / √ h 3 . Set (7.1) b χ = b -hD ξ , ξ -dΓ ε (η) χ(dΓ ε (η)) .
We want to control the error when we consider Lemma 7.7. There exists a constant C independent of χ such that for b χ and u t defined by Equations (7.1) and (7.2),

Tr [b χ ρ app t ] instead of Tr [b χ ρ t ] i.e. we want to control Tr [b χ u t ] with (7.2) u t = ρ t -ρ app t . Since iε∂ t ρ t = [H ε , ρ t ] and iε∂ t ρ app t = [H ε , ρ app t ] -[H ε -H app ε , ρ app t ], the differ- ence u t is solution of the differential equation iε∂ t u t = (ξ -dΓ ε (η)) 2 , u t + Φ ε (f h,ε ), u t -dΓ ε (η) 2 -ε dΓ ε (η 2 ), ρ app
(1)

1 ε ´t 0 Tr b χ (ξ -dΓ ε (η)) 2 , u h,ε,s ds ≤ h ε ´t 0 u h,ε,s L1 ds ≤ C h 2 t 3 ε 3 , (2) 1 ε ´t 0 Tr b χ dΓ ε (η) 2 -ε dΓ ε (η 2 ) , ρ app ds = 0, (3) 1 ε ´t 0 Tr [b χ [Φ ε (f h,ε ) , u s ]] ds ≤ C t 3 h 3/2 ε 7/2 √ ε + t 2 ht ε .
Proof. For Point 1, let us introduce χ 1 ≻ χ (i.e. χ 1 ∈ C ∞ 0 with values in [0, 1] such that χ 1 ≡ 1 on Supp χ) in order to handle only bounded operators: Then we use that both ρt and ρapp t have the same initial value ρ 0 ⊗proj Ω with ρ 0 = j λ j |ψ 0,j ψ 0,j |, j λ j = Tr ρ, λ j ≥ 0, ψ 0,j = 1 to write

Tr b χ (ξ -dΓ ε (η)) 2 , u s = Tr b χ χ 1 (dΓ ε (η))(ξ -dΓ ε (η)) 2 , u s = Tr b χ , χ 1 (dΓ ε (η))(ξ -dΓ ε (η)) 2 u s = Tr χ(dΓ ε (η)) h i {b(x, ξ), ξ 2 }(-hD ξ , ξ -dΓ ε (η)) u s = Tr h i χ(dΓ ε (η)) (2ξ.b)(-hD ξ , ξ -dΓ ε (η)) u s .
ρ t = j λ j |ϕ t,
Tr b χ [Φ ε (f h,ε ) , u s ] ≤ C ∆ Ψs h ε Ĝ L 1 √ ε + s √ 2 f h,ε L 2 ξ .
A time integration gives the result. Proof. We recall that ρ h ∆t = ρ ε ε∆t/h so that with ht ε = ∆t, from Section 7, m b, (ρ χ2 )

  We denote by B(t) (resp. B T (t)) the flow associated with the (resp. dual) Boltzmann equation (2.5), see Section 3. For any non-negative b in C ∞ 0 (R d x × R d * ξ ) we shall prove (1) ´Rd x ×R d * ξ b dµ T ≥ lim inf h→0 Tr[ρ h N,∆t b W (hx, D x )] by the definition of µ T , (2) lim inf h→0 Tr[ρ h N,∆t b W (hx, D x )] ≥ ´Rd x ×R d * ξ (B T (T )b) dµ 0 (see Remark 2.3), (3) ´Rd x ×R d * ξ (B T (T )b) dµ 0 = ´Rd x ×R d * ξ b d(B(T )µ 0 ) by the definition of B(T ).

1

 1 from the mass conservation property of the linear Boltzmann equation and µ T (R dx ×R d ξ ) ≤ 1 from the properties of semiclassical measures. So, necessarily,

  Tr ρ ≤ 1 such that the kernel of ρ = Ad {F x } [ρ] has a bounded support. Introduce the symbol b t = e tQ e 2tξ.∂x b where Q is the collision operator introduced in Equation 3.1 with here σ(ξ, ξ

Proposition 6 . 23 .

 623 The families of operators A (s) = A j µ (s), B(s) = B j µ (s) and C ζ = C j,ζ µ satisfy the hypotheses of Proposition 6.22 with

0 dη= 6 . 5 . 4 .

 0654 iµ1 σ e iσ(P,X) e -µ2is(η .2 -2ξ.η) dη we get the estimate with C A,B = ´Rd η Ĝ(η) |η| dη.For Point 3, the Weyl symbol of ´M 0 B 1 µ (s) e -ζs ds isSymb W eyl ˆM 0 B 1 µ (s) e -ζs ds = ˆRd η Ĝ(η) e µ1iσ e iσ(P,X) e -µ2is(η .2 -2ξ.η)-ζs -µ 2 i (η .2 -2ξ.η)ζ M Symb W eyl C 1,ζ µ + r ζ,M with r ζ,M (x, ξ) = -ˆRd η Ĝ(η) e µ1iσ e iσ(P,X) e -µ2iM(η .2 -2ξ.η)-ζM µ 2 i (η .2 -2ξ.η) + ζ dη .and this expression allows us to get the estimate∂ α x,ξ r ζ,M (x, ξ) ≤ ˆRd η Ĝ(η) P k (M η ) k 1 ζ k+1 e -ζM dη which yields the result with k + 1 replaced by k. The same proof holds for B 2 µ (s) and C 2,ζ µ . Estimate of the error term ∆ -,3 . Proposition 6.24. Let b ∈ C ∞ 0 (R 2d x,ξ ) with Supp ξ b ⊂ B R \ B 1/R for some R > 1. Let γ ∈ (0, 1). There exists a constant C G,b,γ > 0 such that, for all ζ > 0, |∆ -,3 | ≤ ζ γ N k (b)C G,b,γ for some integer k = k (d) big enough.

  By recalling Q ζ -(b) = c ζ b and Q -(b) = c b it is then sufficient to prove Lemma 6.25 below. Lemma 6.25. For any integer k and γ in [0, 1), a positive constant C k,γ,G,C exists such that for ζ ∈ (0, ζ 0 ) sup |α|≤k sup |ξ|∈[R -1 ,R]

For ζ > 0 , 1 π ζ r 2 +ζ 2 .

 012 and r ∈ R, let κ ζ (r) = Lemma 6.30. Let f be a function in the Schwartz class. Then for any γ

7. 2 .

 2 Step 2: Comparison between truncated solutions. Proposition 7.5. Suppose ht ε

t

  with initial data u t=0 = 0. We can then use the integral expressionTr [b χ u t ] = -i ε ˆt 0Tr b χ iε∂ t u t ds . Remark 7.6. Let H be a Hilbert space. If A, B ∈ L(H) and C ∈ L 1 (H), then the cyclicty of the trace gives Tr [A [B , C]] = Tr [[A , B] C].

2 ξ ⊗ΓL 2 η≤ C and a time integra- tion bring 1 ε ˆt 0 ε ˆt 0 u s L1L 2 ξ ⊗ΓL 2 η

 22022 The bound χ(dΓε (η)) (2ξ.b) -hD ξ , ξ -dΓ ε (η) LL Tr b χ (ξ -dΓ ε (η)) 2 , u s ds ≤ C h ds .

  This and the integral above yield the result.For Point 2, let χ 1 ≻ χ,Tr b χ dΓ ε (η) 2ε dΓ ε (η 2 ) , u s = Tr b χ χ 1 (dΓ ε (η)) dΓ ε (η) 2ε dΓ ε (η 2 ) , u s = Tr χ 1 (dΓ ε (η)) dΓ ε (η) 2ε dΓ ε (η 2 ) , b χ u s which vanishes since χ 1 (dΓ ε (η)) dΓ ε (η) 2ε dΓ ε (η 2 ) , b χ = 0 . For Point 3, we have, with ∆ Ψs = Ψs -Ψapp s , Tr b χ [Φ ε (f h,ε ) , u s ] = ∆ Ψs | [b χ , Φ ε (f h,ε )] | Ψs + Ψapp s | [b χ , Φ ε (f h,ε )] |∆ Ψs .Taking the modulus we obtain|Tr [b χ [Φ ε (f h,ε ) , u s ]]| ≤ C ∆ Ψs Φ ε (f h,ε ) Ψs + Φ ε (f h,ε ) b χ Ψs + Φ ε (f h,ε ) b * χ Ψapp s + Φ ε (f h,ε ) Ψapp s and we observe that max Φ ε (f h,ε ) Ψ ♯ s , Φ ε (f h,ε ) b χ Ψ♯ s ≤ C f h,ε (ε + N ε ) 1/2 Ψ♯s and thus, by the number estimate (4) in Proposition 5.3,

7. 3 .Proposition 7 . 8 . 2 . 1 2 1 Ψapp t, 1 |= 1 Ψapp t, 1 | 2 Ψapp t, 2 |≤ 2 Ψapp t, 2 | 2 Ψapp t, 2 |≤ 8 . 8 . 1 .Lemma 8 . 2 . 2 - 1 +

 37821211112222228818221 Step 3: Release of the truncation on the symbol. Let b ∈ C ∞ 0 (R 2dx,ξ ) non-negative, such that Suppξ b ⊂ B R \ B 1/R for some R > 1, ρ ∈ L + 1 L 2 x , Tr ρ ≤ 1, with the support of ρ in B 2 R+1 and χ ∈ C ∞ 0 (R d λ ) with values in [0, 1], χ(B 3R ) = {1}. There is a constant C R,b,χ such that m app (b, ρ, t, χ)m(b, ρ app t ) ≥ E 7.3 with E 7.3 = E 6 + C R,b,χ h, i.e. E 7.3 = C ht ε ht ε + h + h( ht ε ) -1 d/2-1 + h ν(d,α) + h γβ(d,α) + C r,b,χ h .Proof. We restrict the proof to the case of ρ= |ψ ψ| with ψ ∈ L 2 x since ρ is trace class, then ρt = | Ψapp t Ψapp t |. We also define a positive symbol b 1 ∈ C ∞ 0 (R d ξ ) such that Supp b 1 ⊂ [R -2 , R 2 ] and b 1 (ξ 2 ) ≥ b(x, ξ). Then m(b, ρ app t )m app (b, ρ, t, χ) = Tr (1χ(dΓ ε (η))) 1/2 b W (-hD ξ , ξ -dΓ ε (η)) (1χ (dΓ ε (η))) 1/2 ρt ≤ Tr b W 1 (ξ -dΓ ε (η)) .2 1χ(dΓ ε (η)) b W 1 (ξ -dΓ ε (η)) .2 ρt + O(h) with Ψapp t (ξ) = 1 [0,M] (|ξ|) Ψapp t (ξ) and Supp b 1 ⊂ [R -2 , R 2 ]. Then we decompose Ψapp t = 1 [1/2R,2R] (|ξ|) Ψapp t + 1 [0,M]\[1/2R,2R] (|ξ|) Ψapp t = Ψapp t,1 + Ψapp t,With A = b W 1 (ξ -dΓ ε (η)) .2 (1χ(dΓ ε (η))) b W 1 (ξ -dΓ ε (η)).2 ≥ 0 we have the estimateTr A | Ψapp t Ψapp t | ≤ 2 Tr A | Ψapp t,Ψapp t,1 | + 2 Tr A | Ψapp t,Ψapp t,2 | . The first term vanishes since Tr b W 1 (ξ -dΓ ε (η)) .2 1χ(dΓ ε (η)) b W 1 (ξ -dΓ ε (η)) .2 | Ψapp t,Tr 1 [1/2R,2R] (|ξ|) b W 1 (ξ -dΓ ε (η)) .2 (1χ(dΓ ε (η))) b W 1 (ξ -dΓ ε (η)) .2 1 [1/2R,2R] (|ξ|) | Ψapp t,and |ξ| ∈ [1/2R, 2R], |ξ -dΓ ε (η)| ≤ R implies |dΓ ε (η)| ≤ 3R and χ(B 3R ) = {1}.For the second term,Tr b W 1 (ξ -dΓ ε (η)) .2 (1χ(dΓ ε (η))) b W 1 (ξ -dΓ ε (η)) .2 | Ψapp t,Tr b W 1 (ξ -dΓ ε (η)) .2 | Ψapp t,since 1χ(dΓ ε (η)) ≤ Id. Then we use the computation of the evolution of a symbol of |ξ| 2 in the case of the approximated equation as in Remark 6.2 to get that, since b 1 = b 1 (|ξ| 2 ) it is unchanged under the evolution, andTr b W 1 (ξ -dΓ ε (η)) .2 2 | Ψapp t,Tr b W 1 (ξ -dΓ ε (η)) .2 2 | ψ0,2 ⊗ Ω ψ0,2 ⊗ Ω| + E 6which brings the result observing thatTr b W 1 (ξ -dΓ ε (η)) .2 2 | ψ0,2 ⊗ Ω ψ0,2 ⊗ Ω| = Tr b W 1 (ξ .2 ) 2 | ψ0,2 ⊗ Ω ψ0,2 ⊗ Ω|vanishes since Supp b 1 ∩ Supp ψ0,2 = ∅. The derivation of the Boltzmann equation for the model Proposition Let b ∈ C ∞ 0 (R 2d x,ξ ) with Supp ξ b ⊂ B R \ B 1/R . Let ρ a state and T > 0 then lim inf h→0 m B T (T ) b, ρm(b, ρ h N,∆t ) ≤ 0 for a fixed α ∈ ( 3 4 , 1), ∆t = ∆t(h) = h α and N (h) ∆t(h) = T . With b t = e tQ e 2tξ.∂x b, and the hypotheses of Proposition 8.1, m(b ∆t , ρ)m(b, ρ h ∆t ) ≤ C h + (∆t/ √ h) 3 + (∆t/ √ h) 4 + ∆t ∆t + h + (h/∆t) d h µ .

  and from Section 6 also used with ht ε = ∆t we get m(b t , ρ χ2 )m b, (ρ χ2 ) ε,app t ≤ E 6 ≤ E 7 and this term will be in particular controlled if we control the previous one. Finally from the conservation of the support in ξ of the symbol by the approximated Boltzmann equation we getm(b t , ρ)m(b t , ρ χ2 ) ≤ O(h ∞ ) for χ 2 a cutoff function chosen so that χ 2 (B R ) = {1}.Thus we fix, for j = 1, 2, two cutoff functionsχ j ∈ C ∞ 0 (R d λ ) with values in [0, 1], χ j (B Mj ) = {1} for M 1 =3R and M 2 = 1 and with χ 2 (R d \ B R+1 ) = {0}. Proof of Propostition 8.1. Let, for k ∈ N, ∆t > 0, b k,∆t = e ∆tQ e 2∆tξ.∂x k b. Iterating the estimation of the Lemma N (h) times brings m b N,∆t , ρm b, ρ ε N (h),ε∆t/h≤ CN h + (∆t/ √ h) 3 + (∆t/ √ h) 4 + ∆t √ ∆t + h + (h/∆t) d 2 -1 + h µwith N ∆t = T and h α ≤ ht ε = ∆t ≤ 1 for some α ∈ (1/2, 1). Thus we can choose ∆t = th ε = h α and thus N = T h -α . Then we get the estimatem b N,∆t , ρm b, ρ N,ε∆t/h ≤ CT h -α h + h 3α-3/2 + h 4α-2 + h α (h α/2 + h + h (1-α)(d/2-1) + h µ ) ≤ CT o h→0 (1) , for α ∈ ( 3 4,1). Finally it suffices to prove that lim h→0 m b N (h),∆t(h) , ρ = m(b T , ρ) which is true since the estimates of Proposition 3.7 prove that, for some constant C > 0, b N,∆tb T LL 2 x ≤ C N .

  {b, |ξ| 2 } + Qb = 2ξ.∂ x b + Qb .

	Remark 3.2. For a given ξ the integrals in the collision operator only involve the
	values of σ(ξ, |ξ| ω) and b(x, |ξ| ω) for ω ∈ S d-1 .
	We show in Section 3.2 that the dual linear Boltzmann equation is solved by a
	group (B T (t)) t∈R of operators on C 0 ∞ (R d x × R d * ξ ) and in Section 3.3 that it defines by duality a group (B(t)) t∈R of operators on M b (R d x × R d * ξ ).
	3.2. Properties. We recall here the main properties of the dual linear Boltzmann
	equation. (The arguments are the same as for the linear Boltzmann equation.)
	We begin by solving the dual linear Boltzmann equation in L 2 [r, r ′ ] in the sense
	of semigroups.
	Proposition 3.3. Let 0 < r < r ′ < +∞.
	The operator
	• 2ξ.∂ x generates a strongly continuous contraction semigroup on L 2 [r, r ′ ]. • Q is well defined and bounded on H n [r, r ′ ], with

  4) the set of non-negative functions, for t ≥ 0.

	Proof. The properties of generation of groups are clear.
	Point (1) is a consequence of Proposition 3.3.
	Point (2) follows from the Trotter approximation
	B T (t) = lim n→∞	e 2 t n ξ.∂x e	t n Q n ,
	the fact that Q is "local" in (x, |ξ|), and that the speed of propagation of the space-translations is finite when ξ ∈ A ξ [r, r ′ ]. Point (3) follows from (1), (2) and

  2 u 2 which achieves the proof. 4.4. The scaling for field operators. The ε parameter is an intermediate scale which allows to easily identify the graduation in Wick powers. We set Ad{A}[

  j ϕ t,j | , and then u t = j λ j |Ψ t,j -Ψ app

				ρ app t	=	j	λ j |ϕ app t,j	ϕ app t,j | ,
	u t L1L 2 ξ	≤ 2	t,j j	Ψ t,j | -|Ψ app t,j λ j Ψ t,j -Ψ app t,j	Ψ app t,j -Ψ t,j | and ≤ C( ht ε / √ h) 2 .

  + ∆t ∆t + h + (h/∆t) d/2-1 + h µ

	h,app ∆t = m b, (ρ χ2 ) ε,app -m(b, ρ h ∆t ) t -m(b, ρ ε t ) ε / √ h 3 + ht ε / √ h ≤ C h + ht ≤ C h + (∆t/ √ h) 3 + (∆t/ √ h) 4 4 + ht ε	ht ε + h + (ε/t) d/2-1 + h µ

Acknowledgement. We would like to thank Francis Nier and Zied Ammari for very helpful discussions, and many remarks and comments on this paper.

 Notation 6.18. Set (by writing shortly B W for B W (-hD ξ , ξ))

ˆRd Ĝ(η) e µ1iσ τ h P • e -µ2is(η .2 -2ξ.η) dη , (6.8)

µ (s) = ˆRd Ĝ(η) e µ1iσ τ h (px-µ22sη,p ξ ) e -µ2isη .2 dη , (6.9)

px+µ22sη,p ξ ) e µ2isη .2 dη , (6.12)

with σ = σ P, (-2h t ε η, -η) . The terms µ 1 , µ 2 are chosen to adapt to the cases of the terms m ± . More precisely, for j = 1, 2, the previous quantities become

We first show that the operators C ζ µ are good approximations of the operators A µ = ´t/ε 0 A µ (s) ds if the parameter ζ is well chosen. We use the operators ´t/ε 0 B µ (s) ds as an intermediate step. Then we study the limit of the operators C ζ µ , with a distinction between the cases m -and m + .

6.5.2. Estimate of the error terms ∆ ±,1 .

Proposition 6.19. For d ≥ 3,

Proof. The term ∆ ±,1 was defined in Equation (6.4). This inequality follows from Propositions 6.8 and 6.20 below since s → min{1, s -d/2 } is integrable on R + for d ≥ 3.

Proposition 6.20. The families of operators A (s) = A j µ (s) satisfy

Proof. A uniform estimate of Equations (6.8) and (6.11) yields

In order to obtain the part of the estimate with the dependence in s, we use the formula


Panwadee Tangpattanakul
email: panwadee.tangpattanakul@laas.fr

Nicolas Jozefowiez
email: nicolas.jozefowiez@laas.fr

Pierre Lopez
email: pierre.lopez@laas.fr

Multi-Objective Optimization for Selecting and Scheduling Observations by Agile Earth Observing Satellites

Keywords: Multi-objective optimization, Earth observing satellite, scheduling, genetic algorithm

This paper presents a biased random-key genetic algorithm for solving a multi-objective optimization problem concerning the management of agile Earth observing satellites. It addresses the selection and scheduling of a subset of photographs from a set of candidates in order to optimize two objectives: maximizing the total profit, and ensuring fairness among users by minimizing the maximum profit difference between users. Two methods, one based on dominance, the other based on indicator, are compared to select the preferred solutions. The methods are evaluated on realistic instances derived from the 2003 ROADEF challenge.

Introduction

This paper studies the use of multiobjective optimization applied to the scheduling of one Earth observing satellite in a context where multiple users request photographs from the satellite. Genetic algorithms are proposed to solve the problem and experiments are conducted on realistic instances.

The mission of Earth Observing Satellites (EOSs) is to obtain photographs of the Earth surface satisfying users' requirements. When the ground station center receives requests from several users, it has to consider all users' requirements and output an order consisting of a sequence of selected photographs to be transmitted to the satellites. The management problem of EOSs is to select and schedule a subset of photographs from a set of candidates. Among the various types of EOSs, only agile satellites are considered in our study.

An agile EOS has one on-board camera that can move in three axes: roll, pitch, and yaw. It has more efficient capabilities for taking photographs than for example, SPOT5, a non-agile satellite. The selection and scheduling of taking photographs with agile EOSs is more complicated because there are several possible schedules for the same set of selected photographs. The starting time of each photograph is not fixed; nonetheless, it must be within a given time interval. This problem is a scheduling problem and it is the issue under consideration in this paper.

Several algorithms including greedy algorithm, dynamic programming, constraint programming, and local search have been applied for solving agile EOSs scheduling problems [START_REF] Lemaître | Selecting and Scheduling Observations of Agile Satellites[END_REF]. The ROADEF 2003 challenge (see http://challenge. roadef.org/2003/en/) requires the scheduling solutions that maximize total profit of the acquired photographs and also satisfy all physical constraints of agile EOSs. The winner used an algorithm based on simulated annealing [2] and the second prize winner proposed an algorithm based on tabu search [START_REF] Cordeau | Maximizing the Value of an Earth Observation Satellite Orbit[END_REF].

Our work considers agile EOSs scheduling problem where the requests emanate from different users. Hence an objective function to maximize the total profit is not sufficient. The ground station center should also share fairly the resources among users. Therefore, a multi-objective model is considered. The idea to use two objective functions related to fairness and efficiency was proposed in [START_REF] Bataille | Efficiency and Fairness when Sharing the Use of a Satellite[END_REF], and three ways were discussed for solving this sharing problem. The first one gives priority to fairness, the second one to efficiency, and the third one computes a set of trade-offs between fairness and efficiency. For the multi-criteria method, instead of building a complete set of non-dominated solutions, the authors only searched for a decision close to the line with a specified slope on the objective function plane. In [START_REF] Bianchessi | A Heuristic for the Multi-Satellite, Multi-Orbit and Multi-User Management of Earth Observation Satellites[END_REF], a tabu search was used for the multi-satellite, multi-orbit, and multi-user management to select and schedule requests. The upper bounds on the profit were derived by means of a column generation technique. They tested these algorithms with the data instances provided by the French Center for Space Studies (CNES).

This paper proposes a biased random-key genetic algorithm (BRKGA) in order to solve the multi-objective optimization problem for selecting and scheduling the subset of required photographs from multiple users. The two objective functions for this scheduling problem are to maximize the total profit and minimize the maximum difference of profit values between users. The second objective function represents the fairness of resources sharing among the users. The solutions must also satisfy the physical constraints of the agile EOSs.

The article is organized as follows. The problem is explained in Section 2. Section 3 describes the biased random-key genetic algorithm for solving the multi-objective optimization problem. The computational results are reported in Section 4. Finally, conclusions and future work are discussed in Section 5.

Multi-objective optimization for photograph scheduling problem of agile Earth observing satellites

According to the mission and physical constraints of agile EOSs, the requests which are required from users cannot be assigned to a satellite directly. The shape of the area of candidate photographs can be either a spot or polygonal. A spot is a small circular area with a radius of less than 10 km. A polygonal area is an area ranging from 20 to 100 km. All requests (both spot and polygonal area) must be managed from the ground station center by transforming the requests into rectangular shapes called strips for which the camera can take a photograph at once. Each spot is considered as one single strip. Each polygonal area is decomposed into several strips with fixed width but variable length. Each strip can be acquired following two possible opposite directions as shown in Figure 1, only one of them will be selected in the scheduling results. Requests can be mono or stereo photographs. A mono photograph is taken only once, whereas a stereo photograph must be acquired twice in the same direction but from different angles. In [START_REF] Verfaillie | Management of the Mission of Earth Observation Satellites Challenge Description[END_REF], a simplified version of the problem of managing the mission of agile Earth observing satellites was presented. An instance gives the set of candidate requests with shape type, mono or stereo characteristic, associated gain and surface areas. Let r be the set of requests. These requests are divided into the set of strips s. Each strip includes details, which consist of the identity of request R[j] where that strip is split from, the useful surface area Su[j], duration time Du[j], and earliest and latest visible times from two ends T e[j, 0], T l[j, 0], T e[j, 1], and T l[j, 1]. Each strip is possibly taken from two directions but only one can be selected. Thus, our scheduling problem is solved for selecting and scheduling the possible strip acquisition that is associated with the possible acquisition direction of each strip. If one possible strip acquisition is selected, the other one (possible acquisition in opposite direction) of the same strip is forbidden to be selected. For the profit calculation of each acquired request, its profit can be calculated by a piecewise linear function of gain depending on the fraction of taken useful area and the whole area of each request, as illustrated in Figure 2.

Hence, we extend the case to multiple users as in [START_REF] Bianchessi | A Heuristic for the Multi-Satellite, Multi-Orbit and Multi-User Management of Earth Observation Satellites[END_REF]. However, we solve the problem as a real bi-objective problem. The two objectives are to maximize total profit and ensure fairness between users. For the second objective, the defined function is to minimize the maximum difference in profit between the users. The Fig. 2. Piecewise linear function of gain P (x) depending on the effective ratio x of acquired area [START_REF] Verfaillie | Management of the Mission of Earth Observation Satellites Challenge Description[END_REF] imperative constraints for finding the feasible solutions are: take each strip within their associated time windows, no overlapping images, sufficient transition times, acquire only one direction for each strip, and satisfy the stereoscopic constraint for stereo requests.

3 Biased random-key genetic algorithm for the multi-user photograph scheduling

Genetic algorithm is a heuristic search method that mimics the process of natural evolution. The starting step of genetic algorithm is the initial population generation and the population consists of several chromosomes. Each chromosome, which is formed of several genes, represents one solution. The genetic algorithm involves three mechanisms (selection, crossover, and mutation) to generate the new chromosomes for the next generation and repeats to generate the new generation until the stopping criterion is satisfied. We propose a genetic algorithm for selecting and scheduling the required photographs for the agile EOSs from multi-user requests. The biased random-key genetic algorithm (BRKGA) [START_REF] Gonçalves | Biased Random-Key Genetic Algorithms for Combinatorial Optimization[END_REF] is used to solve this scheduling problem with two important steps (encoding and decoding). Two methods are used to select the preferred solutions in each genetic algorithm iteration: i) fast nondominated sorting with crowding distance assignment [START_REF] Deb | A Fast and Elite Multiobjective Genetic Algorithm: NSGA-II[END_REF]; ii) indicator based on the hypervolume concept [START_REF] Zitzler | Indicator-Based Selection in Multiobjective Search[END_REF]. Let p, p e , and p m be the sizes of the population, of the elite set, and of the mutation set, respectively.

Chromosome generation in the encoding process

The initial population consisting of p chromosomes is generated. Each chromosome consists of genes which are encoded by real values randomly generated in the interval (0, 1]. For our problem, each strip can be taken following two opposite directions (but only one direction will be selected). Each gene is associated with one direction of a strip, which is a possible strip acquisition that we will just call acquisition in the sequel, for the sake of simplification. It is the reason why we define the number of genes to be equal to twice the number of strips.

Schedule generation in the decoding process

Each chromosome is decoded in order to obtain one solution, which is a sequence of selected acquisitions of the scheduling problem. In this decoding step, the considered priority of each acquisition depends on the gene values: from high to low. The imperative constraints, except the stereo constraint, are verified during each considered acquisition. The stereo constraint is checked once all acquisitions have been treated. All constraints must be satisfied in order to obtain a feasible solution. The flowchart of these decoding steps is depicted in Figure 3.

The example of one solution from the modified instance, which needs to schedule two strips, is shown in Figure 4. Two strips are considered in this instance; therefore the number of genes equals 4. The random-keys are generated for all genes and each gene represents one acquisition. The decoding steps are used to obtain the sequence of selected acquisitions and the values of the two objective functions.

Biased random-key genetic algorithm

In BRKGA, the new population is combined from three parts (selection, crossover, and mutation) [START_REF] Gonçalves | Biased Random-Key Genetic Algorithms for Combinatorial Optimization[END_REF]. The first part is the selection part in which we can choose a selection method from several efficient algorithms, e.g., NSGA-II [START_REF] Deb | A Fast and Elite Multiobjective Genetic Algorithm: NSGA-II[END_REF], IBEA [START_REF] Zitzler | Indicator-Based Selection in Multiobjective Search[END_REF], SMS-EMOA [START_REF] Beume | SMS-EMOA: Multiobjective selection based on dominated hypervolume[END_REF], etc. We propose two selection methods to choose p e preferred chromosomes (elite set) from the current population. We copy these p e chromosomes to the top part of the next population. The two methods are:

Fast nondominated sorting and crowding distance assignment

Fast nondominated sorting and crowding distance assignment methods were proposed in the Nondominated Sorting Genetic Algorithm II (NSGA-II) [START_REF] Deb | A Fast and Elite Multiobjective Genetic Algorithm: NSGA-II[END_REF].

In our work, the fast non-dominated sorting method is used to find the solutions in rank zero (nondominated solutions). If the number of nondominated solutions is more than the parameter setting value of maximum size of the elite set, the crowding distance assignment method is applied to select some solutions from the nondominated set to become the elite set. Otherwise all nondominated solutions will become the elite set. The concept of the crowding distance assignment method is to get an estimate of the density of solutions surrounding a particular solution in the population.

Indicator based on the hypervolume concept

The use of an indicator based on the hypervolume concept was proposed in the Indicator-Based Evolutionary Algorithm (IBEA) [START_REF] Zitzler | Indicator-Based Selection in Multiobjective Search[END_REF]. The indicator based method is used to assign fitness values based on the hypervolume concept to the population members and some solutions in the current population are selected to become the elite set for next population. The indicator-based method performs binary tournaments for all solutions in the current population and implements environmental selection by removing the worst solution from the population and by updating the fitness values of the remaining solutions. The worst solution is removed repeatedly until the number of remaining solutions satisfies the recommended size of the elite set for BRKGA.

The second part is the bottom part which is the mutant set. It is the set of p m chromosomes generated to avoid entrapment in a local optimum. These chromosomes are randomly generated by the same method used to generate the initial population. The last part is the crossover part for which each crossover offspring is built from one elite chromosome and one chromosome in the previous population. Each element in the crossover offspring is obtained from the element in elite chromosome with the probability ρ e . The crossover offspring is stored in the middle part of the new population. Hence, the size of crossover offspring set is p -p e -p m to fulfill the remaining space of chromosomes in the next population. The process for generating the next populations is applied repeatedly until the stopping criterion is satisfied.

Computational results

The ROADEF 2003 challenge instances (subsetA) from ROADEF Challenge website (http://challenge.roadef.org/2003/en/sujet.php) are modified for 4-user requirements and the format of instance names are changed to a b c, where a is the number of requests, b is the number of stereo requests, and c is the number of strips. For the proposed biased random-key genetic algorithm, the recommended parameter value settings is displayed in Table 1 [START_REF] Gonçalves | Biased Random-Key Genetic Algorithms for Combinatorial Optimization[END_REF]. Two population sizes of n and 2n, where n is the length of a chromosome, are tested. The best solutions are stored in the archive set. If there is at least one solution from the current population that can dominate some solutions in the archive set, the archive set will be updated. Thus, we use the number of iterations of the last archive set improvement to be the stopping criterion. The algorithms were experimentally tuned and the stopping value is set to 50. The size of the elite set is equal to the number of non-repeating photograph scheduling results from the nondominated solutions, but it is not over 0.15p. The size of the mutant set is 0.3p. The probability of elite element inheritance for crossover operation is 0.6. Two methods (dominance-based and indicator-based) for selecting some solutions to become the elite set are tested. They are implemented in C++. We test ten runs per instance. Hypervolumes of the approximate Pareto front are calculated by using a reference point of 0 for the first objective and the maximum of the profit summations of each user for the second one. The average and standard deviations values of hypervolumes, the average number of solutions, and average CPU times are reported in Table 2. Each method obtains a set of solutions, which considers both objective functions (maximize total profit and ensure fairness among users) and all constraints of agile EOSs are satisfied. For both methods, when comparing the results from two different population sizes, most of them show that the methods with the population size 2n obtain the better average and standard deviation values of hypervolumes and acquire more solutions, but CPU times are higher. In the other way, when we compare results between dominance-based and indicator-based, the average values of hypervolumes cannot show exactly which one obtains the better solutions. However, the standard deviation for the population size 2n of indicator-based is better than dominance-based. On the number of solutions and CPU time, dominance-based obtains more solutions and spends less CPU times, especially for large instances. Except for instance 12 2 25, dominance-based takes very high CPU time and this is strange. Hence, more tests are done to check the number of iterations until the stopping criterion is satisfied for instance 12 2 25 and instance 12 9 28. The average number of iterations for instance 12 2 25 and instance 12 9 28 are 2657535.7 and 177.7, respectively. Therefore, instance 12 2 25 spends very high CPU time, because it uses a huge number of iterations until the stopping criterion is satisfied. For instance 2 0 2 when using the population size n, both methods cannot reach any result, because the population size is too small for generating the new generation from 3 parts in BRKGA. Nevertheless, the computation times for large instances are quite high, that means that the efficiency of the decoding methods certainly deserves to be improved.

Conclusions and future work

Multi-objective optimization is applied to solve the problem of selecting and scheduling the observations of agile Earth observing satellites. The instances of ROADEF 2003 challenge are modified in order to take account explicitly of 4-user requirements. Two objective functions are considered to maximize the total profit and to minimize the maximum difference profit between users for the fairness of resource sharing. Moreover, all constraints have to be satisfied. A biased random-key genetic algorithm (BRKGA) is applied to solve this problem. Random-key encoding generates each chromosome in the population and all of them are decoded to be the solutions. Thus, two methods, fast nondominated sorting with crowding distance assignment on the one hand and indicator based on the hypervolume concept on the other hand, are used for selecting the elite set of solutions from the population. An elite set, a crossover offsprings set, and a mutant set are combined to become the next population. The results of the dominance-based and indicator-based methods with two population sizes are compared. The approximate solutions are obtained but the computation times for large instances are quite high.

This work is still in progress. As a future work, we plan to use other randomkey decoding methods in order to reduce the computation times. Moreover, we will apply indicator-based multi-objective local search (IBMOLS) to solve this problem and compare the IBMOLS results with the BRKGA results which are proposed in this paper.

Fig. 1 .

 1 Fig.1. A polygonal area is decomposed into several strips; each strip can be acquired according to two possible directions

Fig. 3 .Fig. 4 .

 34 Fig. 3. Decoding steps flowchart of one chromosome into one solution

Table 1 .

 1 Recommended parameter values of BRKGA[START_REF] Gonçalves | Biased Random-Key Genetic Algorithms for Combinatorial Optimization[END_REF]

	Parameter Recommended value
	p	p = a.n,
		where 1 ≤ a ∈ R is a constant and
		n is the length of the chromosome
	pe	0.10p ≤ pe ≤ 0.25p
	pm	0.10p ≤ pm ≤ 0.30p
	ρe	0.5 ≤ ρe ≤ 0.8

Acknowledgment

This work was supported by THEOS Operational Training Programme (TOTP). The authors gratefully acknowledge this support. Thanks are also due to the referees for their valuable comments.


Brigitte Juanals
email: brigitte.juanals@orange.fr

Martin Lafréchoux

Jean-Luc Minel

Modyco

The Standardization Process: Exploring the Battlefield

In the domain of the international standardization of social security, the organization standardscalled management standards-of security are published (or are in the process of being published) by NGOs authorized to conduct standardization activities. These texts deal with the security and protection of people and goods; they are characterized by their international, European or national production process and are widely applied in a transnational space to both private and public organizations.

The specificity of our research field -information security management standards in the broader international context of social security -led us to design an analytical framework capable of describing the texts of standards in order to apprehend the communicational and organizational policies recommended by these texts. This framework must also be relevant to deal with the process of the construction of terminology which is carried out by the technical committees (TC) in charge of this task. Our research aims to identify, in a specific industrial branch, the conditions and mechanisms which entail the introduction of new standards and how these mechanisms result in the crystallization, communication and circulation of recommended practices, knowledge and procedures, means of organization and socio-technical arrangements.

Our research covers several themes: i) What are the motivations and the communication strategies of organizations which are the "authors-players" in the field? ii) What kind of mechanisms are organizations using to reference these standards and how do they put them into practice? iii) In the texts of the standards themselves, where do the concepts originate from and how are they developed, what are the diachronic variants of terms used to lexicalize them? iv) What kind of software should be developed to study the networks of influence that seek to impose a given terminology and hence the concepts used by the standards? Our paper will focus on point 4, as research on the other themes has already been published (or submitted) in [START_REF] Juanals | Writing and Monitoring in International Standardization, Theoretical Choices and Methodological Tools[END_REF][START_REF] Juanals | Les acteurs et les textes de la normalisation internationale de la sécurité sociétale. Construction d'une approche communicationnelle[END_REF], Lafréchoux &al. (2012a), andLafréchoux &[START_REF] Juanals | Pour une approche communicationnelle des normes internationales de management de la sécurité sociétale[END_REF]. We will present the building of a new tool, called KONTRAST, which aims to combine a socio-organizational analysis with a linguistic one. More precisely, we try to leverage the terminological variants in a non-stabilized terminological domain and to track the lobbying and influences at work in the battlefield [START_REF] Lafréchoux | KONTRAST: a Terminological Ontological Glossary to Exploit Linguistic Variants[END_REF].