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...OUHOT 5. Hard spheres 40 6. Extensions and complements 52 References 58 hal-00447988, version 1 - 18 Jan 2010 1. Introduction and main results 1.1. The Boltzmann equation. The Boltzmann equation (Cf. =-=[10]-=- and [11]) describes the behavior of a dilute gas when the only interactions taken into account are binary collisions. It writes ∂f (1.1) ∂t +v ·∇xf = Q(f,f) where Q = Q(f,f) is the bilinear Boltzmann...

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...Hard spheres 40 6. Extensions and complements 52 References 58 hal-00447988, version 1 - 18 Jan 2010 1. Introduction and main results 1.1. The Boltzmann equation. The Boltzmann equation (Cf. [10] and =-=[11]-=-) describes the behavior of a dilute gas when the only interactions taken into account are binary collisions. It writes ∂f (1.1) ∂t +v ·∇xf = Q(f,f) where Q = Q(f,f) is the bilinear Boltzmann collisio...

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...olecules but not hard spheres.ON CHAOS PROPAGATION OF BOLTZMANN COLLISION PROCESSES 5 hal-00447988, version 1 - 18 Jan 2010 A completely different approach was undertaken by Sznitman in the eighties =-=[41]-=- (see also Tanaka [43]). Starting from the observation that Grünbaum’s proof was incomplete, he gave a full proof of chaos propagation for hard spheres. His work was based on: (1) a new uniqueness res...

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...lled 6-th Hilbert problem mentionned by Hilbert at the International Congress of Mathematics at Paris in 1900. At least at the formal level, the correct limiting procedure has been identified by Grad =-=[18]-=- in the late fourties (see also [9] for mathematical formulation of the open question): it is now called the Boltzmann-Grad or low density limit. However the original question of Hilbert remains large...

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...in C([0,∞);P(PG1 )) to (6.41) starting from π0. Moreover, if π0 is f0-chaotic, then πt is SNL t f0-chaotic for any t ≥ 0. Proof of Theorem 6.8. Step 1: Chaos propagation. From Hewitt-Savage’s theorem =-=[21]-=-, for any π ∈ P(P(E)) there exists a unique sequence (π ℓ ) ∈ P(E ℓ ) such that the identities 〈π ℓ ,ϕ〉 = = ∫ ∫ P(E) 〈f ⊗ℓ ,ϕ〉π(df) R P(E) ℓ ϕ (f)π(df) = 〈π,Rℓ ϕ 〉, hal-00447988, version 1 - 18 Jan 20...

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...10 hal-00447988, version 1 - 18 Jan 2010 Abstract. This paper is devoted to the study of mean-field limit for systems of indistinguables particles undergoing collision processes. As formulated by Kac =-=[22]-=- this limit is based on the chaos propagation, and we (1) prove and quantify this property for Boltzmann collision processes with unbounded collision rates (hard spheres or longrange interactions), (2...

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...tion of the open question): it is now called the Boltzmann-Grad or low density limit. However the original question of Hilbert remains largely open, in spite of a striking breakthrough due to Lanford =-=[25]-=-, who proved the limit for short times. The tremendous difficulty underlying this limit is the irreversibility of the Boltzmann equation, whereas the particle system interacting via Newton’s laws is a...

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...t and gt to the Boltzmann equation for Maxwellian collision kernel satisfy (4.9) sup|ft −gt| 2 ≤ |f0 −g0| 2 . t≥0 Proof of Lemma 4.3. We recall Bobylev’s identity for Maxwellian collision kernel (cf. =-=[4]-=-) F ( Q + (f,g) ) (ξ) = ˆ Q + (F,G)(ξ) =: 1 ∫ ( b σ · 2 ˆ ) ξ [F + G − +F − G + ]dσ, S d−1 with F = ˆ f, G = ˆg, F ± = F(ξ ± ), G ± = G(ξ ± ), ˆ ξ = ξ/|ξ| and ξ + = 1 2 (ξ +|ξ|σ), ξ− = 1 (ξ −|ξ|σ). 2 ...

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...[34]). It turns out that it lead us to develop a new theory, inspiring from more recent tools such as the course of Lions on “Mean-field games” at Collège de France, and the master courses of Méléard =-=[33]-=- and Villani [45] on mean-field limits. One of the byproduct of our paper is that we make fully rigorous the original intuition of Grünbaum in order to prove chaos propagation for the Boltzmann veloci...

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... k2,k3 > 0, the spaces PGk and PGk are topologically 2 3 uniformly equivalent on bounded sets of PGk . 1 Example 2.7. There are many distances on P(E) which induce the weak topology, see for instance =-=[37]-=-. In section 2.7 below, we will present some of them which have a practical interest for us, and which are all topologically uniformly equivalent on “bounded sets” of P(E), when the bounded sets are d...

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...ion generates a nonlinear semigroup SNL t (f0) := ft for any f0 ∈ P2(R d ) (probabilities with bounded second moment): for the Maxwell case we refer to [42, 44], for the hard spheres case we refer to =-=[35]-=- in a L 1 setting, and for the generalization of these L 1 solutions to P2(R d ) we refer to [14], [17] and [28]. For these solutions, one has the conservation of momentum and energy ∫ ∫ ∀t ≥ 0, vft(d...

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... distances are all topologically equivalent to the weak topology σ(P(E),Cb(E)) (on the sets BPk,a(E) for k large enough and for any a ∈ (0,∞)) and they are all uniformly topologically equivalent (see =-=[44, 8]-=- and section 2.8). Example 2.18 (Dual-Hölder (or Zolotarev’s) distances). Denote by distE a distance on E and let us fix v0 ∈ E (e.g. v0 = 0 when E = Rd in the sequel). Denote by C 0,s 0 (E), s ∈ (0,1...

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...ion is defined by (1.2), (1.3), (1.4). The equation generates a nonlinear semigroup SNL t (f0) := ft for any f0 ∈ P2(R d ) (probabilities with bounded second moment): for the Maxwell case we refer to =-=[42, 44]-=-, for the hard spheres case we refer to [35] in a L 1 setting, and for the generalization of these L 1 solutions to P2(R d ) we refer to [14], [17] and [28]. For these solutions, one has the conservat...

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...−g0| η 2 . As a consequence, the operator L NL M 1 (G1,r,G1,r). t defined by L NL t [f0](h0) := ht is such that L NL t [f0] ∈ Proof of Lemma 4.5. Step 1. Estimate in |·|4. We proceed in the spirit of =-=[42, 7]-=-. With the notation M = M4, ˆ M = ˆ M4, introduced in Example 2.21, let us define d := f −g, s := f +g and ˜ d := d−M[d], and then D := F(d), S := F(s) and ˜D := F( ˜ d) = D − ˆM[d]. hal-00447988, ver...

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...inequality (5.31) implies (possibly increasing the constant K) Wt ≤ W0 +K ∫ t 0 Ws (logWs) − ds. From the Gronwall lemma we deduce (5.32) Wt ≤ (W0) exp(−Kt) whenever Wt ≤ 1/2. On the other hand, from =-=[5]-=- and [36], there exists λ2,Z > 0, z ∈ (0,¯z) such that ∀t ≥ 0 ‖ft −Mf0 ‖ L 1 mz +‖gt −Mg0 ‖ L 1 mz ≤ Ze−λt , where Mf0 and Mg0 stand again for the normalized Maxwellian associated to f0 and g0. Denoti...

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...sup‖ζt‖M 1 ≤ e 2 [0,T] C (1+t)√ max{Mk(f0),Mk(g0)} ‖f0 −g0‖ 2−5/k M1 , from which estimate (5.6) follows. 5.7. Proof of (A4) uniformly in time. Let us start from an auxiliary result. It was proved in =-=[36]-=- that the nonlinear and linearized Boltzmann semigroups for hard spheres satisfy (5.21) ∥ ∥ NL S t ∥ L1 −1 (mz ) ≤ Cze −λt , ∥ Lt e ∥ −λt L1 −1 ≤ Cze (mz ) where mz(v) := e z|v| , z > 0, λ = λ(E) is t...

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... distances are all topologically equivalent to the weak topology σ(P(E),Cb(E)) (on the sets BPk,a(E) for k large enough and for any a ∈ (0,∞)) and they are all uniformly topologically equivalent (see =-=[44, 8]-=- and section 2.8). Example 2.18 (Dual-Hölder (or Zolotarev’s) distances). Denote by distE a distance on E and let us fix v0 ∈ E (e.g. v0 = 0 when E = Rd in the sequel). Denote by C 0,s 0 (E), s ∈ (0,1...

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...ariables on (R d ) N is then built by iterating the above construction. After scaling the time (changing t → t/N in order that the number of interactions is of order O(1) on finite time interval, see =-=[39]-=-) we denote by fN t the law of Vt, SN t the associated semigroup, GN and TN t respectively the dual generator and dual semigroup, as in the previous abstract construction. The so-called Master equatio...

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...icle) system has trajectories includedin thechaotic ones. Hence themethod of Sznitman proves convergence but does not provide any rate for chaoticity. Let us also emphasize that Graham and Méléard in =-=[19]-=- have obtained a rate of convergence (of order 1/ √ N) on any bounded finite interval of the N-particles system to the deterministic Boltzmann dynamic in the case of Maxwell molecules under Grad’s cut...

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...with the one of a Lévy process and the first quantitative chaos propagation result for the spatially homogeneous Boltzmann equation for hard spheres (improvement of the convergence result of Sznitman =-=[40]-=-). Moreover our chaos propagation results are the first uniform in time ones for Boltzmann collision processes (to our knowledge), which partly answers the important question raised by Kac of relating...

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...ions around the deterministic limit along time). (3) Proof of propagation in time of the convergence to a “universal behavior” around the deterministic limit (central limit theorem). See for instance =-=[31, 38]-=- for related results. (4) Proof of propagation in time of bounds of exponential type on the “rare” events far from chaoticity (large deviation estimates). (5) Estimations (2)–(4) can be made uniformly...

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... based on two “unproved assumptions on the Boltzmann flow” (page 328): (a) existence and uniqueness for measure solutions and (b) a smoothness assumption. Assumption (a) was indeed recently proved in =-=[17]-=- using Wasserstein metrics techniques andin [14] adaptingtheclassical DiBlasio trick [13], butconcerningassumption (b), although it was inspired by cutoff maxwell molecules (for which it is true), it ...

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...ever, thanks to Povzner’s inequality, one may show that ‖ft‖ M 1 3 ∈ L 1 (0,T) whenever ‖f0‖ M 1 2,1 is finite, with the definition ‖f0‖ M 1 k,ℓ := ∫ 〈v〉 k log(〈v〉) ℓ df0(v) < +∞ (see (5.13) below or =-=[35, 27]-=-), which will be the key step for establishing (5.4) and (5.5). Now, our goal is to estimate (in terms of ‖g0 −f0‖ M 1) the M 1 norm of ζt := ft −gt −ht. The measure ζt satisfies the following evoluti...

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...nd more generally unbounded collision kernels, although his method seemed impossible to extend (no semi-explicit combinatorial formula of the solution exists in this case). In the seventies, Grünbaum =-=[20]-=- then proposed in a very compact and abstract paper another method for dealing with hard spheres, based on the Trotter-Kato formula for semigroups and a clever functional framework (partially remindfu...

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...uniqueness for measure solutions and (b) a smoothness assumption. Assumption (a) was indeed recently proved in [17] using Wasserstein metrics techniques andin [14] adaptingtheclassical DiBlasio trick =-=[13]-=-, butconcerningassumption (b), although it was inspired by cutoff maxwell molecules (for which it is true), it fails for hard spheres (cf. the counterexample built by Lu and Wennberg in [29]) and is s...

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...√ N) on any bounded finite interval of the N-particles system to the deterministic Boltzmann dynamic in the case of Maxwell molecules under Grad’s cut-off hypothesis, and that Fournier and Méléard in =-=[15, 16]-=- have obtained the convergence of the Monte-Carlo approximation (with numerical cutoff) of the Boltzmann equation for true Maxwell molecules with a rate of convergence (depending on the numerical cuto...

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... point (1) in the case of his baby one-dimensional model. The key point in his analysis is a clever combinatorial use of a semi-explicit form of the solution (Wild sums). It was generalized by McKean =-=[32]-=- to the Boltzmann collision operator but only for “Maxwell molecules with cutoff”, i.e., roughly when the collision kernel B above is constant. In this case the combinatorial argument of Kac can be ex...

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...∈ [1,∞), one has ( N (2.22) Wq µ V ,µ N Y and that ) = dℓ q (E N /SN)(V,Y) := min σ∈SN (2.23) ∀f, g ∈ P1(E), W1(f,g) = [f −g] ∗ 1 ( 1 N N∑ i=1 distE(vi,y σ(i)) q = sup 〈f −g,φ〉. φ∈Lip0(E) We refer to =-=[46]-=- and the references therein for more details on the Wasserstein distances. )1/q , Example 2.20 (Fourier-based norms). For E = R d , mG1 := |v|, mG1 := 0, let us define ∀f ∈ TPG1 , ‖f‖G1 = |f|s := sup ...

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...sio trick [13], butconcerningassumption (b), although it was inspired by cutoff maxwell molecules (for which it is true), it fails for hard spheres (cf. the counterexample built by Lu and Wennberg in =-=[29]-=-) and is somehow “too rough” in this case. (2) A key part in the proof in this paper is the expansion of the “Hf” function, which is an a clever idea of Grünbaum (and the starting point for our idea o...

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...the one hand, for any π0 ∈ P(PG1 ) we may definethe semigroup (St) on P(PG1 ) and the flow (¯πt) by setting ¯πt = S∞ t π0 and (duality formula) ∀Φ ∈ Cb(PG1 ;R) 〈S∞ t π0,Φ〉 = 〈π0,T ∞ t Φ〉. Remark (see =-=[3]-=-) that S∞ t π0 ∈ (Cb(P(V)) ′ + = P(P(E)). Under the additional assumption that PG1,a is compact for any a (that is true in our application cases when ‖ · ‖G1 metrizes the weak measures topology) that...

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... P.-L. Lions recently developed in his course at Collège de France [26] or the one developed by L. Ambrosio et al in order to deal with gradient flows in probability measures spaces, see for instance =-=[2]-=-. In the sequel we develop a differential calculus in probability measures spaces into a simple and robust framework, well suited to deal with the different objects we have to manipulate (1-particle s...

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... by Hilbert at the International Congress of Mathematics at Paris in 1900. At least at the formal level, the correct limiting procedure has been identified by Grad [18] in the late fourties (see also =-=[9]-=- for mathematical formulation of the open question): it is now called the Boltzmann-Grad or low density limit. However the original question of Hilbert remains largely open, in spite of a striking bre...

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...√ N) on any bounded finite interval of the N-particles system to the deterministic Boltzmann dynamic in the case of Maxwell molecules under Grad’s cut-off hypothesis, and that Fournier and Méléard in =-=[15, 16]-=- have obtained the convergence of the Monte-Carlo approximation (with numerical cutoff) of the Boltzmann equation for true Maxwell molecules with a rate of convergence (depending on the numerical cuto...

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... P(E) (or subsets of P(E)) as “plunged sub-manifolds” of some larger normed spaces Gi. Our approach thus differs from the approach of P.-L. Lions recently developed in his course at Collège de France =-=[26]-=- or the one developed by L. Ambrosio et al in order to deal with gradient flows in probability measures spaces, see for instance [2]. In the sequel we develop a differential calculus in probability me...

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...s f-chaotic, for a given probability f ∈ P(E), if for any ℓ ∈ N ∗ and any ϕ ∈ Cb(E) ⊗ℓ there holds lim N→∞ 〈 f N ,ϕ⊗1 N−ℓ〉 = 〈 f ⊗ℓ ,ϕ which amounts to the weak convergence of any marginals (see also =-=[6]-=- for another stronger notion of “entropic” chaoticity). Here we will deal with quantified chaoticity, in the sense that we measure precisely the rate of convergence in the above limit. Namely, we say ...

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...triction, by using the change of variable σ ↦→ −σ, from now on we restrict the angular domain to θ ∈ [−π/2,π/2] for the limiting equation as well as the N-particle system. Therefore we assume suppb ⊂ =-=[0,1]-=-. We still denote by b the symmetrized version of b by a slight abuse of notation. 4.2. Statement of the result. In this section we consider the case of the Maxwell molecules kernel. More precisely we...

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... √ θg0 |2|ξ| 2 dξ 〈ξ〉 2s ( |θf0 −θg0 |η +|uf0 −ug0 |2η) ( |Ef0 −Eg0 |η +|uf0 −ug0 |2η) ( W2(f0,g0) 2η +W1(f0,g0) 2η) ( W1(f0,g0) η/2 +W1(f0,g0) 2η) ) η hal-00447988, version 1 - 18 Jan 2010 (see also =-=[12]-=- for more general estimates of the Wasserstein distance between two gaussians). Gathering these two estimates, we deduce (5.33) ∀t ≥ 0 Wt ≤ Z1e −λt +Z2W η/2 0 . Let us (implicitly) define T, ¯ W0 ∈ (0...

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...mann flow” (page 328): (a) existence and uniqueness for measure solutions and (b) a smoothness assumption. Assumption (a) was indeed recently proved in [17] using Wasserstein metrics techniques andin =-=[14]-=- adaptingtheclassical DiBlasio trick [13], butconcerningassumption (b), although it was inspired by cutoff maxwell molecules (for which it is true), it fails for hard spheres (cf. the counterexample b...

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...ment): for the Maxwell case we refer to [42, 44], for the hard spheres case we refer to [35] in a L 1 setting, and for the generalization of these L 1 solutions to P2(R d ) we refer to [14], [17] and =-=[28]-=-. For these solutions, one has the conservation of momentum and energy ∫ ∫ ∀t ≥ 0, vft(dv) = vf0(dv), R d R d ∫ R d |v| 2 ∫ ft(dv) = R d |v| 2 f0(dv). Without restriction, by using the change of varia...

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...knowledgments: The authors would like to thank the mathematics departement of Chalmers University for the invitation in november 2008, where the abstract method was devised and the related joint work =-=[34]-=- with Bernt Wennberg was initiated. The second author would like to thank Cambridge University who provided repeated hospitality in 2009 thanks to the Award No. KUK-I1-007-43, funded by the King Abdul...

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...ions around the deterministic limit along time). (3) Proof of propagation in time of the convergence to a “universal behavior” around the deterministic limit (central limit theorem). See for instance =-=[31, 38]-=- for related results. (4) Proof of propagation in time of bounds of exponential type on the “rare” events far from chaoticity (large deviation estimates). (5) Estimations (2)–(4) can be made uniformly...

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...spheres.ON CHAOS PROPAGATION OF BOLTZMANN COLLISION PROCESSES 5 hal-00447988, version 1 - 18 Jan 2010 A completely different approach was undertaken by Sznitman in the eighties [41] (see also Tanaka =-=[43]-=-). Starting from the observation that Grünbaum’s proof was incomplete, he gave a full proof of chaos propagation for hard spheres. His work was based on: (1) a new uniqueness result for measures for t...