Quantitative and qualitative Kac’s chaos on the Boltzmann’s sphere

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www.imstat.org/aihp 2015, Vol. 51, No. 3, 993–1039

DOI:10.1214/14-AIHP612

Quantitative and qualitative Kac’s chaos on the Boltzmann’s sphere

Kleber Carrapatoso

Ceremade, Université Paris Dauphine, Place du Maréchal De Lattre De Tassigny, 75775 Paris cedex 16, France.

Received 22 June 2012; revised 6 June 2013; accepted 5 March 2014

Abstract. We investigate the construction of chaotic probability measures on the Boltzmann’s sphere, which is the state space of the stochastic process of a many-particle system undergoing a dynamics preserving energy and momentum.

Firstly, based on a version of the local Central Limit Theorem (or Berry–Esseen theorem), we construct a sequence of probabilities that is Kac chaotic and we prove a quantitative rate of convergence. Then, we investigate a stronger notion of chaos, namely entropic chaos introduced in (Kinet. Relat. Models3(2010) 85–122), and we prove, with quantitative rate, that this same sequence is also entropically chaotic.

Furthermore, we investigate more general class of probability measures on the Boltzmann’s sphere. Using the HWI inequality we prove that a Kac chaotic probability with bounded Fisher’s information is entropically chaotic and we give a quantitative rate.

We also link different notions of chaos, proving that Fisher’s information chaos, introduced in (On Kac’s chaos and related problems (2012) Preprint), is stronger than entropic chaos, which is stronger than Kac’s chaos. We give a possible answer to (Kinet. Relat.

Models3(2010) 85–122), Open Problem 11, in the Boltzmann’s sphere’s framework.

Finally, applying our previous results to the recent results on propagation of chaos for the Boltzmann equation (Invent. Math.

193(2013) 1–147), we prove a quantitative rate for the propagation of entropic chaos for the Boltzmann equation with Maxwellian molecules.

Résumé. Nous étudions la construction de mesures de probabilité chaotiques sur la sphère de Boltzmann, qui est l’espace d’état du processus stochastique d’un système de particules subissant une dynamique en préservant l’énergie et la quantité de mouvement.

Premièrement, basé sur une version du Théorème Central Limite (ou théorème de Berry–Esseen) locale, nous construisons une suite de probabilités qui est chaotique au sens de Kac et nous prouvons un taux quantitatif de convergence. Ensuite, nous étudions une notion plus forte de chaos, le chaos entropique introduit dans (Kinet. Relat. Models3(2010) 85–122), et nous prouvons, avec un taux quantitatif, que cette même suite est également entropie chaotique.

Par ailleurs, nous nous intéressons à des classes plus générale de mesures de probabilité sur la sphère de Boltzmann. En utilisant l’inégalité HWI nous montrons qu’une probabilité chaotique au sens de Kac qui possède l’information de Fisher bornée est entropie chaotique et nous donnons un taux quantitatif. Nous relions également les différentes notions de chaos, en montrant que le Fisher chaos, introduit dans (On Kac’s chaos and related problems (2012) Preprint), est plus fort que le chaos entropique, qui est plus fort que le chaos au sens de Kac. Nous donnons une réponse possible à (Kinet. Relat. Models3(2010) 85–122), Open Problem 11, dans le cadre de la sphère de Boltzmann.

Finalement, en appliquant nos résultats précédents aux résultats récents sur la propagation du chaos pour l’équation de Boltz- mann (Invent. Math.193 (2013) 1–147), nous démontrons un taux quantitatif pour la propagation du chaos entropique pour l’équation de Boltzmann avec des molécules maxwelliennes.

MSC:76P05; 60G50; 54C70; 82C40

Keywords:Kac’s chaos; Entropic chaos; Fisher’s information chaos; Many-particle jump process; Entropy; Fisher’s information; Mean-field limit;

Central Limit Theorem; Berry–Esseen; HWI inequality; Boltzmann equation

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1. Introduction 1.1. Motivation

In his celebrated paper [9], Kac introduced the notion of propagation of chaos in order to connect a stochastic process of a system ofN identical particles undergoing binary collisions to its mean field equation.

Our interest in this paper is to investigate chaotic distributions supported by the phase space of the stochastic process of theN-particle system as we shall explain. We refer to [3] for a detailed introduction on this topic and on Kac’s paper [9].

Consider a system ofNidentical particles of massρ >0 such that its evolution is described by a jump process with binary collisions that preserves energy and momentum. Let us denote byi, j the particles undergoing the collision, with pre-collisional velocitiesv_i, vj∈R^d and post-collisional velocitiesv^∗_i, v_j^∗∈R^d. We have then the conservation of momentum

ρv_i^∗+ρv_j^∗=ρv_i+ρv_j, and the conservation of energy

ρ

2v_i^∗²+ρ

2v_j^∗²=ρ

2|v_i|²+ρ 2|v_j|². If the system has initial energyE=¹₂N

i=1ρ|vi|²∈R+and initial momentumM=ρm=N

i=1ρvi∈R^d, then both energy and momentum will be unchanged under the dynamics. The phase space of this process is then the manifoldS^N(√

E, m)⊂R^dNdefined by S^N(√

E, m):=

V =(v1, . . . , v_N)∈R^dN1 2

N i=1

ρ|v_i|²=E, N i=1

ρv_i=ρm

, which is the intersection of a sphere of radius√

2E/ρand a hyperplane. This spaceS^N(√

E, m)is in fact a sphere in R^dNof dimensiond(N−1)−1 with radius

2E/ρ− |m|²/Nand center(m, . . . , m)/√

N. We remark that we need

|m|²≤2NE/ρin order toS^N(√

E, m)be non empty.

Now choosing units such that the massρof each particle is equal to 2, the total value of kinetic energy isdNand, without loss of generality, choosingm=0, the state space of this dynamics is

S_B^N:=S^N(√

dN ,0)=

V =(v1, . . . , vN)∈R^dN N i=1

|vi|²=dN, N i=1

vi=0

(1) and we shall call the manifoldS_B^Nthe Boltzmann’s sphere.

An example of this kind of dynamics is the space homogeneous Boltzmann model that we shall explain. Given a pre-collisional system of velocitiesV =(v1, . . . , vN)∈R^dN and a collision kernel (for more information on the collision kernel we refer to [13,17])

B(z,cosθ )=Γ

|z| b(cosθ ), (2)

for some nonnegative functionsΓ andb, the process is:

• for anyi=j, pick a random timeT (Γ (|v_i−v_j|))of collision accordingly to an exponential law of parameter Γ (|v_i−v_j|)and choose the minimum timeT1and the colliding pair(vi, vj)such that

T₁=T Γ

|v_i−v_j|

=min

i,jT Γ

|v_i−v_j| ,

• drawσ∈S^d⁻¹⊂R^daccording to the lawb(cosθ_ij), with cosθ_ij=σ·(vi−vj)

|v_i−v_j|,

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• after collision the new velocities become V_ij^∗=

v₁, . . . , v^∗_i, . . . , v_j^∗, . . . , v_N ,

where the post-collisional velocitiesv^∗_i andv^∗_j are given by v^∗_i =v_i+v_j

2 +|v_i−v_j|

2 σ, v_j^∗=v_i+v_j

2 −|v_i−v_j|

2 σ. (3)

Iterating this construction we built then the associated Markov process(Vt)_t_≥₀onR^dN. The equation of the associated law is given by, after a rescaling of time (see [13]),

∂_tG^N_t =L_NG^N_t = 1 N

i<j

S^d⁻¹

G^N_t

V_ij^∗ −G^N_t (V ) B

|v_i−v_j|,cosθ dσ (4)

with initial dataG^N₀ and whereV_ij^∗=(v₁, . . . , v_i^∗, . . . , v_j^∗, . . . , v_N). This equation is known as themaster equation.

Associated to this process, we have the (limit) spatially homogeneous Boltzmann equation [13,14,17]

∂_tf (t, v)=

R^d×S^d⁻¹B

|v−w|,cosθ f w^∗ f

v^∗ −f (w)f (v) dwdσ (5)

with initial dataf (0,·)=f₀and where the post-collisional velocitiesv^∗andw^∗are obtained by (3).

We shall highlight here the models we consider in the last part of this work (see Theorem8below), and we refer to [17] for more details concerning the collision kernel. Assuming a collision kernelB derived from inverse-power law interaction potentials

φ(r)=r⁻^(s⁻¹⁾, s >2,

we have that the collision kernel has the form B(z,cosθ )= |z|^γb(cosθ ), γ =s−(2d−1)

s−1 , (6)

where the functionbis locally smooth and has a nonintegrable singularity

sin^d⁻²θ b(cosθ )∼θ∼0C_bθ⁻¹⁻^ν, ν∈(0,2), Cb>0. (7) In the particular case of three dimensionsd=3, we haveγ=(s−5)/(s−1)andν=2/(s−1). If we replace the angular collision kernelbby a locally integrable one, we speak of cutoff collision kernels (or Grad’s cutoff).

We shall consider in this work the case of Maxwellian molecules, in which the collision kernel does not depend on the relative velocity, i.e.γ=0 in (6). We consider the general assumption

B(|v−w|,cosθ )=b(cosθ ),

∀α >0, _π

0 b(cosθ )(1−cosθ )^α⁺^1/4sin^d⁻²θdθ <+∞. (8) This is the same assumption made in [13], since in Theorem8we use their results. Remark that (8) includes the true Maxwellian molecules (or Maxwellian molecules without cutoff) in dimensiond=3, whenγ=0,ν=1/2 and

B(z,cosθ )=b(cosθ ), b(cosθ )∼θ∼0C_bθ⁻^5/2 (d=3). (9) Also, it includes the Grad’s cutoff Maxwellian molecules, when the singularity is removed,

B(z,cosθ )=b(cosθ ),

_π

0

b(cosθ )sin^d⁻²θdθ <+∞. (10)

Some results in Theorem8will consider the general assumption (8) and others the cutoff Maxwellian molecules (10).

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The program set by Kac in [9] was to investigate the behavior of solutions of the mean field equation (5) in terms of the behaviour of the solutions of the master equation (4). Moreover, the notion of propagation of chaos introduced by Kac means that if the initial distributionG^N₀ isf₀-chaotic (Definition1below) then, for allt >0, the solutionG^N_t of (4) isf_t-chaotic, wheref_t is the solution of (5). For more information on this topic we refer to the recent results of Mischler, Mouhot and Wennberg [13,14].

This paper is inspired by the works of Carlen, Carvalho, Le Roux, Loss and Villani [3] and also of Hauray and Mischler [8], which investigate chaotic probabilities on the usual sphere inR^N with radius√

N (also called Kac’s sphere). This sphere is the phase space of Kac’s model, which is a one-dimensional simplification, introduced in [9], of the model presented above, with energy conservation only.

The novelty here is that we investigate chaotic probability sequences in the Boltzmann’s sphereS_B^N⊂R^dN and, furthermore, we prove quantitative rates of chaos convergence. Moreover, we apply our results to the Boltzmann equation with true Maxwellian molecules to prove quantitative propagation of entropic chaos.

1.2. Definitions and main results

LetEbe a Polish space, then we shall denote byP(E)the space of Borel probability measures onE. Furthermore, through this paper, on the spaceE^N we will only consider symmetric measures, more precisely, we say thatG^N∈ P(E^N)is symmetric if for allϕ∈Cb(E^N)we have

E^N

ϕdG^N=

E^N

ϕ_σdG^N,

for any permutationσ of{1, . . . , N}, and where ϕσ:=ϕ(Vσ)=ϕ(vσ (1), . . . , vσ (N )), forV =(v₁, . . . , v_N)∈E^N.

ForG^N∈P(E^N)and a integer∈ [1, N]we denote byG^N (orΠ(G^N)) the-marginal ofG^N, defined by

∀ϕ∈C_b E ,

E

ϕdG^N =

E^N

ϕ⊗1^⊗^(N⁻⁾dG^N.

We shall use through the paper the same notation to represent a probability measure and its density with respect to the Lebesgue measure.

We can now give the notion of chaos formalized by Kac in [9], we also refer to [16] for an introduction on this topic with a probabilistic approach and to [12] for a short survey.

Definition 1 (Kac’s chaos). Considerf ∈P(E).We say thatG^N∈P(E^N)isf-chaotic(or f-Kac chaotic),if for each fixed positive integer,G^N converges tof^⊗in the sense of measures inP(E)whenN goes to infinity,i.e.if for allϕ∈Cb(E),

Nlim→∞

E

ϕdG^N =

E

ϕdf^⊗. (11)

In fact, it is well known that we need condition (11) to hold for only one≥2 (see for instance [16]).

We also introduce the Monge–Kantorovich–Wasserstein (MKW) distance and for more information about it we refer to [18]. Consider an integerandp∈ [1,∞), we define then the space

Pp

E :=

F∈P

E ;M_p F :=

E

|X|^pdF(X) <∞

.

Then, forF, G∈Pp(E)we define the MKW distance betweenFandGby W_p

F, G := inf

π∈Π (F,G)

E×E

d_E(X, Y )^pdπ(X, Y ) 1/p

, (12)

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whereΠ (F, G)is the set of transfer plan betweenFandG, which is the set of probabilty measures onE×E with marginalsFandGrespectively, and where we define the distaced_Eas

∀X=(x₁, . . . , x), Y=(y₁, . . . , y)∈E, d_E(X, Y ):=

i=1

d_E(x_i, y_i).

In the paper we will use the Euclidean distance inE=R^d, i.e.d_E(x_i, y_i)= |x_i−y_i|for allx_i, y_i∈E. More precisely, we shall use

∀f, g∈P1

R^d , W₁(f, g)= inf

π∈Π (f,g)

R^d×R^d|x−y|dπ(x, y) and

∀f, g∈P2

R^d , W₂(f, g)= inf

π∈Π (f,g)

R^d×R^d|x−y|²dπ(x, y) 1/2

.

Moreover, forF^N, G^N∈P(S_B^N)we shall use in the definition ofW_p(F^N, G^N)the Euclidean distance inherited from R^dN, which means that forX, Y∈S_B^Nwe shall used_SN

B(X, Y )= |X−Y|.

Letγbe the Gaussian probability measure onR^d,γ (v)=(2π)⁻^d/2e^−|^v^|²^/2, andμ∈P(R^d). We define the relative entropy ofμwith respect toγ by

H (μ|γ ):=

R^dlogdμ

dγ dμ, (13)

ifμis absolutely continuous with respect toγ, otherwiseH (μ|γ ):= +∞.

Moreover, forG^N∈P(S_B^N)we define the relative entropy with respect toγ^N, the uniform probability measure onS_B^N, by

H

G^N|γ^N :=

S_B^N

logdG^N dγ^N

dG^N (14)

shall now define a stronger notion of chaos, namely the entropic chaos introduced in [3].

Definition 2 (Entropic chaos). We say that the sequenceG^N∈P(S_B^N)is entropicallyf-chaotic,for somef ∈P(R^d), ifG^Nisf-chaotic in Kac’s sense(Definition1)and

Nlim→∞

1 NH

G^N|γ^N =H (f|γ ) (15)

withH (f|γ ) <∞.

Finally, with these definitions at hand we can state the main results of the paper.

Theorem 3. For anyf ∈P6(R^d)∩L^p(R^d)with1< p≤ ∞,there exists a sequence of probability measuresF^N:=

[f^⊗^N]_S^N

B ∈P(S_B^N),contructed by conditioning theN-fold tensorization off to the Boltzmann’s sphere,such that (i) F^Nisf-chaotic.More precisely,for any≥1fixed there exists a constantC=C() >0such that forN≥+1

we have W₁

F^N, f^⊗ ≤ C

√N;

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(ii) F^N is entropicallyf-chaotic.More precisely,there exists a constantC >0such that 1

NH

F^N|γ^N −H (f|γ ) ≤ C

√N.

Let us now define the relative Fisher’s information of a probability measureμ∈P(R^d)with respect toγby I (μ|γ ):=

R^d

∇logdμ dγ

²dμ, (16)

and, as we did for entropy, we also define forG^N∈P(S_B^N)the relative Fisher’s information with respect toγ^Nby I

G^N|γ^N :=

S_B^N

∇_SlogdG^N dγ^N

²dG^N, (17)

where∇_S stands for the gradient on the Boltzmann’s sphere, i.e. the component of the usual gradient inR^dN that is tangent to the sphereS_B^N.

We define then another stronger notion of chaos, the Fisher’s information chaos, in an analogous way of Defini- tion2.

Definition 4 (Fisher’s information chaos). We say that the sequenceG^N∈P(S_B^N)is Fisher’s informationf-chaotic, for somef∈P(R^d),ifG^Nisf-chaotic in Kac’s sense(Definition1)and

Nlim→∞

1 NI

G^N|γ^N =I (f|γ ) withI (f|γ ) <∞.

Remark 5. The Fisher’s information chaos is introduced in[8]in a weaker way,which is in fact equivalent to Defini- tion4thanks to Theorem6.

Next, we may compare as follows the several notions of chaos:

Theorem 6. ConsiderG^N∈P(S_B^N),withkth order momentMk(G^N₁)bounded,for some k≥6,and suppose that G^N₁ f inP(R^d).

Then,each assertion listed below implies the further one:

(i) N⁻¹I (G^N|γ^N)→I (f|γ ),withI (f|γ ) <∞.

(ii) N⁻¹I (G^N|γ^N)is bounded andG^Nisf-chaotic in Kac’s sense.

(iii) N⁻¹H (G^N|γ^N)→H (f|γ ),withH (f|γ ) <∞.

(iv) G^Nisf-chaotic in Kac’s sense.

As a consequence, in Definition2of the entropic chaos and in Definition4of Fisher’s information chaos, we only need the convergence of the first marginal, i.e.G^N₁ f, instead of the convergence of all marginals. Hence, this theorem asserts that Fisher’s information chaos implies entropic chaos, which in turns implies chaos (or Kac’s chaos).

Furthermore, we prove a quantitative rate for the implication (ii)⇒(iii).

Another main result of the paper is a possible answer to [3], Open Problem 11, in the setting of Boltzmann’s sphere given in Theorem7. First of all, let us state the problem. ForG^N∈P(S_B^N)andf ∈P6(R^d)∩L^p(R^d)withp >1, consider the following two conditions:

Nlim→∞

1 NH

G^N| f^⊗^N

S^N_B =0, (18)

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and

∀∈N, lim

N→∞H

G^N |f^⊗ =0, (19)

where[f^⊗^N]_S^N

B is the probability measure constructed in Theorem3. In the Kac’s sphere setting (i.e.S^N⁻¹(√ N ) instead ofS_B^N), [3] proved that condition (19) holds whenG^N is the conditioned tensor productG^N= [f^⊗^N]_S^N

B. As discussed in [3], conditions (15), (18) and (19) really mean thatG^N is “strongly” close tof^⊗^N, not only in the weak measure sense for marginals as in Kac’s chaos. In view of this, they formulated the following problem.

Problem 1 ([3], Open Problem 11). Does condition(18)imply condition(19)? More generally,does condition(19) hold for a larger and easily recognized class of chaotic sequences,larger than those contructed by means of condi- tioning tensor products?

We give a partial answer to Problem1in the following theorem.

Theorem 7. ConsiderG^N∈P(S_B^N)such thatG^N isf-chaotic,for somef ∈P(R^d),and suppose that Mk

G^N₁ ≤C, k >2, 1 NI

G^N|γ^N ≤C.

Suppose further that f ∈L^∞(R^d)andf (v₁)≥exp(−a|v₁|²)for some constanta >0.Then for any fixed,there exists a constantC=C(d, ,fL^∞, Mk(G^N₁, f )) >0such that for allN≥+1we have

H

G^N|f^⊗ ≤CW₁

G^N , f^⊗ ^{θ (,d,k)},

whereθ (, d, k) is constructive and depends on,d andk.As a consequence,H (G^N |f^⊗)→0 asN → ∞and condition(19)holds.

This theorem exhibes a class of chaotic sequences in the Boltzmann’s sphere that satisfy condition (19). At a first sight, the hypotheses needed onG^Nandf to (19) be true may seen stronger than the conditioned tensor product, in which case [3] proved that (19) holds (as said above). However, as remarked in [3,8], the conditioned tensor product assumption is not propagated along time by the Boltzmann equation but the assumptions needed in Theorem7may be.

It is indeed true for the Boltzmann equation with Maxwellian molecules (see point (iv) of Theorem8below for a precise statement), hence, in this setting, the assumptions in Theorem7are natural, which gives a satisfying answer to the second question on Problem1in the Maxwellian case.

The interest here is that, as already remarked in [3,8,13], a natural step on Kac’s program would be to study the propagation of conditions (15) or (18) or (19) (which are stronger than Kac’s chaos) under the master equation (4).

As explained above, as a consequence of Theorem7, the propagation of (19) holds true for Maxwellian molecules.

We continue the investigation of these issues in Theorem8below, proving also the propagation of entropic chaos (15) and (18).

We can apply our previous results to the Boltzmann equation for Maxwellian molecules. Some of the results concern assumption (8), i.e. Maxwellian molecules with and without cutoff, others concern only the Grad’s cutoff Maxwellian molecules (10). Thanks to the work on propagation of chaos of [13], we can establish the following theorem.

Theorem 8. Letf0∈P(R^d)andG^N₀ ∈P(S_B^N).Consider then,for allt >0,the solutionG^N_t of the Boltzmann master equation(4)with Maxellian molecules((8)or(10))associated to the initial conditionG^N₀,and the solutionf_t of the limiting Boltzmann equation(5)with Maxellian molecules((8)or(10))associated to the initial dataf0.

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Then we have

(i) Let(10)be in force.Considerf₀∈P6∩L^p(R^d)forp >1.IfG^N₀ is entropicallyf₀-chaotic,then for allt >0, G^N_t is entropicallyft-chaotic,more precisely

Nlim→∞

1 NH

G^N_t |γ^N =H (f_t|γ ).

(ii) Let(8)be in force.Considerf₀∈P6(R^d)withI (f₀|γ ) <∞.IfG^N₀ = [f₀^⊗^N]_S^N

B ∈P(S_B^N)as in Theorem3,then, for allt >0,G^N_t is entropicallyf_t-chaotic.More precisely,for any

< 48

(7d+6)²(5d+24)

there exists a constantC:=C() >0such that sup

t≥0

1 NH

G^N_t |γ^N −H (f_t|γ )

≤CN⁻.

(iii) Let(10)be in force.Considerf0∈P6∩L^∞(R^d)andf0(v1)≥exp(−α|v1|²+β)forα >0andβ∈R.IfG^N₀ satisfies condition(18)

Nlim→∞

1 NH

G^N₀| f₀^⊗^N

S^N_B =0,

then,for allt >0,G^N_t also satisfies condition(18)

Nlim→∞

1 NH

G^N_t | f_t^⊗^N

S^N_B =0.

(iv) Let(10)be in force.Considerf0∈P6∩L^∞(R^d)andf0(v1)≥exp(−α|v1|²+β)forα >0,β∈R.Consider alsoG^N₀ that isf₀-chaotic and hasM_k(Π₁(G^N₀))andN⁻¹I (G^N₀|γ^N)finite,for somek >2.

Then,for allt≥0,G^N_t satisfies condition(19)

∀∈N, lim

N→∞H Π

G^N_t |f_t^⊗ =0.

Theorem8improves the results of [13] where Kac’s chaos is established with a rate but entropic chaos is proved without any rate. Indeed, point (i) here is proved in [13] and point (ii) gives a quantitative propagation of entropic chaos. Moreover, point (iii) answers a question of [13], Remark 7.11, and point (iv) is a consequence of Theorem7as said above.

It is worth mentioning that point (i) was proved in [13] for both the Maxwellian molecules with cutoff (10) and the hard spheres case (which corresponds to the collision kernelB(z,cosθ )= |z|). The proof of point (iii) also shows that (iii) is valid for hard spheres, indeed the proof is based on the fact that (15) and (18) are equivalent under some hypotheses on f (see Theorem25) and these properties are also propagated along time in the hard spheres case (propagation ofL^∞, moments and lower Maxwellian bounds, see e.g. [17] and the references therein). However, the results (ii) and (iv) are valid only for the Maxwellian case, the reason behind this is that a key ingredient of the proof is the propagation of the Fisher’s information bound, and such property is only know to hold for Maxwellian molecules.

1.3. Strategy

We construct a probability onS_B^N based on tensorization and conditioning of some probabilty measure onR^d. To this purpose, we use an explicit formula for the marginals of the uniform probablity onS_B^Nand a version of the local Central Limit Theorem (also known as Berry–Esseen), which is the cornerstone of the proof.

In order to study more general probabilities on the Boltzmann’s sphere, we use an interpolation-type inequality, relating entropy, Fisher’s information and the 2-MKW distance, called HWI inequality from [10,15,18], to show that Kac chaotic probabilities with finite Fisher’s information are entropically chaotic.

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Finally, the application of our results to the Boltzmann equation is based on recent results of propagation of chaos from [13] and on the relations of different notions of measuring chaos from the work [8].

1.4. Previous works

In [9] it is proved that theN-fold tensorization of a smooth probability onRconditioned to the Kac’s sphere, i.e. the usual sphereS^N⁻¹(√

N ), is Kac chaotic. Then, the work [3] extends this result to a more general class of probabilities onR, introduces the notion of entropic chaos and also proves that theN-fold tensorization conditioned to the Kac’s sphere is entropically chaotic. Furthermore, the recent work [8] gives quantitative rates of the results before, introduces the notion of Fisher’s information chaos and links these three notions of chaos.

1.5. Organization of the paper

In Section2we shall study the uniform probability measure onS_B^N. In Section3we construct a chaotic distribution on Boltzmann’s sphere based on a probability measure onR^d. Furthermore we prove a quantitative chaos convergence rate and we prove point (i) of Theorem3. Then, in Section4we investigate the entropic and Fisher’s information chaos.

First, we study the entropic chaos for the probability distribution built before in Section3and we prove point (ii) of Theorem3. Then, we link these three notions of chaos and investigate a more general class of probability measures onS_B^N, proving Theorem6and Theorem7. Finally, in Section5we use our previous results to prove Theorem8.

2. Uniform probability measure

ConsiderV =(v₁, . . . , v_N)∈R^dN,r∈R₊andz∈R^d. We define the sphere S^N(r, z):=

V =(v₁, . . . , v_N)∈R^dN N i=1

v_i²=r², N

i=1

v_i=z

.

We denote by γ_r,z^N the uniform probability measure on S^N(r, z). We recall that S_B^N :=S^N(√

dN ,0) is the Boltzmann sphere and we denote by γ^N:=γ√^N

dN ,0 its uniform probability measure. Moreover, we also denote by Sⁿ⁻¹(r)⊂Rⁿthe usual sphere of dimensionn−1 and radiusr,Sⁿ⁻¹:=Sⁿ⁻¹(1)and by|Sⁿ⁻¹|its measure. We can easily compute the measure ofS^N(r, z)by

S^N(r, z)=S^d(N⁻¹⁾⁻¹

r²−|z|² N

(d(N−1)−1)/2 +

. (20)

For V =(v₁, . . . , v_N)∈ R^dN, we shall use through the paper the notation V =(v₁, . . . , v)∈R^d, V_,N = (v₊₁, . . . , v_N)∈R^d(N⁻⁾andV¯=

i=1v_i∈R^d.

We begin with the following result of a change of variables, proved in AppendixA.1.

Lemma 9. ConsiderV ∈S^N(r, z).We can make a change of coordinates(v1, . . . , vN)→(u1, . . . , uN)in the follow- ing way

u_N= 1

√N(v₁+ · · · +v_N),

(21) uk= 1

√k(k+1)(v1+ · · · +vk−kvk+1), 1≤k≤N−1,

such that the Jacobian is equal to one,|u1|²+ · · · + |u_N|²= |v1|²+ · · · + |v_N|²and |v1|²+ · · · + |v_N|²=r²,

v1,α+ · · · +vN,α=zα →

|u₁|²+ · · · + |u_N₋₁|²=r²−^|^z_N^|²,

u_N,α=^√^z^α_N, 1≤α≤d. (22)

(10)

With these definitions and notations at hand we can study some properties of the uniform probability measureγ^N onS_B^N. We remark that these estimates can also be obtained using correlation operators on the Boltzmann’s sphere as in Carlen, Carvalho and Loss [4].

Lemma 10. We have the following properties

(i) for any≤N−1the-marginal ofγ^Nis given byγ^N(dV)=γ^N(V)dVwith γ^N(V)=|S^d(N⁻⁻¹⁾⁻¹|

|S^d(N⁻¹⁾⁻¹|

N^d/2 (N−)^d/2

(dN− |V|²− | ¯V|²/(N−))^(d(N₊ ⁻⁻¹⁾⁻^2)/2

(dN )^(d(N⁻¹⁾⁻^2)/2 , (23)

wheredV=dv1· · ·dvis the Lebesgue measure onR^d.

(ii) the moments ofγ^N are uniformly bounded in N,more precisely,for k≥1 we haveMk(γ^N)≤Cd,k,,where C_d,k,depends ond, kand.

Before the proof, we refer to [7] where a Fubini-like theorem onS^N(r, z)is proved, which yields a generalization of (23) for the-marginal ofγ_r,z^N.

Proof of Lemma10. Let us split the proof.

(i) We can defineγ_r,z^N by γ_r,z^N:= 1

Z_r,z^N lim

h→0

1 h(1_BN

z(r+h)−1_BN

z(r)), B_z^N(r):=

V ∈R^dN; |V| ≤r, N i=1

v_i=z

, whereZ^N_r,zis the normalization constant so that the integral ofγ_r,z^N is one.

Considerϕ∈C(R^d), for≤N−1, then 1_BN

z(r), ϕ⊗1^N⁻

=

R^dN1_|_V_|2+|V,N|²≤r²1_V_¯

+v₊₁+···+v_N=zϕ(V)dVdV,N

=

R^dϕ(V)

R^d(N⁻⁾1_|_V_,N_|2≤r²−|V|²1_v₊₁_+···+_v

N=z− ¯VdV,N

dV

=

R^dϕ(V)B^d(N⁻⁻¹⁾

r²− |V|²−|z− ¯V|² N−

d(N−−1)/2

+ dV,

where|B^d(N⁻⁻¹⁾|is the measure of the unit ball in dimensiond(N−−1). We deduce then that the-marginal of γ_r,z^N, denoted byΠ(γ_r,z^N), is given by

Π

γ_r,z^N = 1 Z_r,z^N

d dr

B^d(N⁻⁻¹⁾

r²− |V|²−|z− ¯V|² N−

d(N−−1)/2 +

=|B^d(N⁻⁻¹⁾|

Z^N_r,z d(N−−1)r

r²− |V|²−|z− ¯V|² N−

(d(N−−1)−2)/2 +

=|S^d(N⁻⁻¹⁾⁻¹| Z_r,z^N r

r²− |V|²−|z− ¯V|² N−

(d(N−−1)−2)/2 +

and in the particular caser²=dN,z=0 Π

γ^N =γ^N=|S^d(N⁻⁻¹⁾⁻¹| Z√^N

dN ,0

(dN )^1/2

dN− |V|²− | ¯V|² N−

(d(N−−1)−2)/2 +

. (24)

(11)

Now we shall computeZ^N:=Z√^N

dN ,0, with Z^N=S^d(N⁻⁻¹⁾⁻¹(dN )^1/2

R^d

dN− |V|²− | ¯V|² N−

(d(N−−1)−2)/2 +

dV. (25)

We start by the integral A=

R^d

dN− |V|²− | ¯V|² N−

(d(N−−1)−2)/2 +

dV,

with the changement of variable (21)–(22) (replacingN by), with the notation U=U₋₁=(u₁, . . . , u₋₁)and x=uto simplify, we obtain

A=

R^d

dN− |U|²− N N−|x|²

(d(N−−1)−2)/2 +

dUdx.

ChangingUto spherical coordinates in dimensiond(−1), we have A=

R^d

_∞

0

S^d(⁻¹⁾⁻¹

dN−ρ²− N N−|x|²

(d(N−−1)−2)/2 +

ρ^d(⁻¹⁾⁻¹dρdx

=S^d(⁻¹⁾⁻¹ _∞

0

R^d

(d(N−−1)−2)/2

+ dx

ρ^d(⁻¹⁾⁻¹dρ. (26) Looking first to the integral overR^dwe obtain, changingxto spherical coordinates in dimensiond,

B=

R^d

(d(N−−1)−2)/2 +

dx

=S^d⁻¹ _∞

0

dN−ρ²− N N−y²

(d(N−−1)−2)/2 +

y^d⁻¹dy, and after some computations we get

B=|S^d⁻¹| 2

N− N

d/2

dN−ρ² ^(d(N₊ ⁻⁾⁻^2)/2 ₁

0

(1−y)^(d(N⁻⁻¹⁾⁻^2)/2y^(d⁻^2)/2dy

=|S^d⁻¹| 2

N− N

d/2

dN−ρ² ^(d(N₊ ⁻⁾⁻^2)/2Γ ((d(N−−1)−2)/2+1)Γ ((d−2)/2+1) Γ ((d(N−−1)−2)/2+(d−2)/2+2) . Plugging this expression in (26) we get

A=S^d(⁻¹⁾⁻¹|S^d⁻¹| 2

N− N

d/2Γ ((d(N−−1)−2)/2+1)Γ ((d−2)/2+1) Γ ((d(N−−1)−2)/2+(d−2)/2+2)

× _∞

0

dN−ρ² ^(d(N₊ ⁻⁾⁻^2)/2ρ^d(⁻¹⁾⁻¹dρ,

and we can compute the last integral C:=

_∞

0

dN−ρ² ^(d(N₊ ⁻⁾⁻^2)/2ρ^d(⁻¹⁾⁻¹dρ

= 1

2(dN )^(d(N⁻¹⁾⁻^2)/2Γ ((d(N−)−2)/2+1)Γ ((d(−1)−2)/2+1) Γ ((d(N−)−2)/2+(d(−1)−2)/2+2) .