Cedric Villani - Cercignani's inequality (Fields Medal 2010)

CERCIGNANI’S CONJECTURE IS SOMETIMES TRUEAND ALWAYS ALMOST TRUE

CEDRIC VILLANI

Abstract. We establish several new functional inequalities comparing Boltz-mann’s entropy production functional with the relative H functional. First weprove a longstanding conjecture by Cercignani under the nonphysical assumptionthat the Boltzmann collision kernel is superquadratic at infinity. The proof restson the method introduced in [39] combined with a novel use of the Blachman-Stam inequality. If the superquadraticity assumption is not satisfied, then it isknown that Cercignani’s conjecture is not true; however we establish a slightlyweakened version of it for all physically relevant collision kernels, thus extendingprevious results from [39]. Finally, we consider the entropy-entropy productionversion of Kac’s spectral gap problem and obtain estimates about the dependenceof the constants with respect to the dimension. The first two results are sharpin some sense, and the third one is likely to be, too; they contain all previouslyknown entropy estimates as particular cases. This gives a first coherent picture ofthe study of entropy production, according to a program started by Carlen andCarvalho [12] ten years ago. These entropy inequalities are one step in our studyof the trend to equilibrium for the Boltzmann equation.

Contents

1. Introduction 22. Superquadratic collision kernels 103. Nonvanishing collision kernels 174. General collision kernels 205. Further developments and open problems 306. The entropy variant of Kac’s problem 32References 41

Date: January 31, 2005.

1

2 CEDRIC VILLANI

1. Introduction

Cercignani’s conjecture [19] asserts the domination of Boltzmann’s relative Hfunctional by a constant multiple of Boltzmann’s entropy production functional.Since its formulation twenty years ago, it has been disproved in greater and greatergenerality [8, 10, 47]. Nevertheless, we shall show in this paper that it is true incertain cases. In fact, if Grad’s angular cut-off is imposed, then it holds true inessentially all the cases which were not previously covered by counterexamples. Weshall also show that a slightly weaker family of inequalities holds true in all physicalcases, thus recovering and improving previous results in this direction [12, 13, 39].These functional inequalities play a key role in our subsequent treatment of trend toequilibrium for the Boltzmann equation, both in a spatially homogeneous [34] andin a spatially inhomogeneous context [25]. We will not develop these issues here, inorder to limit the size of the present paper; the reader who would like to consult atentative global view on the subject is referred to [42, chapter C], or, better, to [46].

Before going further, let us give precise definitions of all the quantities which willbe under study. Whenever f = f(v) is a nonnegative integrable function on R

N

(N ≥ 2), to be thought of as a density in velocity space, we define

1) the macroscopic density, velocity and temperature associated to f , by theidentities

(1) ρ =

∫

RN

f(v) dv; u =1

ρ

∫

RN

f(v)v dv; T =1

Nρ

∫

RN

f(v)|v − u|2 dv;

2) the H functional, or negative of the entropy, by

(2) H(f) =

∫

RN

f log f ;

3) the thermodynamical equilibrium, or maximum of the entropy under theconstraints (1),

(3) M f (v) = Mρ,u,T (v) ≡ ρ e−|v−u|2

2T

(2πT )N/2;

CERCIGNANI’S CONJECTURE IS SOMETIMES TRUE 3

4) the H-dissipation, or entropy production1:

(4) D(f) =1

4

∫

RN×RN×SN−1

(f ′f ′∗ − ff∗) log

f ′f ′∗

ff∗B(v − v∗, σ) dσ dv dv∗.

Here SN−1 stands for the unit sphere in RN ; we have used the shorthands f ′ = f(v′),

f∗ = f(v∗), f′∗ = f(v′∗), where

(5) v′ =v + v∗

2+|v − v∗|

2σ, v′∗ =

v + v∗2− |v − v∗|

2σ (σ ∈ SN−1).

Finally, the function B appearing in (4) is Boltzmann’s collision kernel2.

Boltzmann’s entropy production functional describes the amount of entropy whichis produced per unit of time by the collisions of particles in a dilute gas, at agiven position in space. These collisions are assumed to be elastic and binary; onemay think of v′, v′∗ as the respective velocities of particles which are just about tocollide, and will have respective velocities v, v∗ as a result of this interaction. As forBoltzmann’s collision kernel, it depends on the particular interaction between theparticles, but it is always assumed to depend only on the two parameters |v − v∗|(modulus of the relative velocity) and cos θ = 〈k, σ〉 (cosine of the deviation angle),where k = (v− v∗)/|v− v∗|. Typical examples are the hard-sphere models, in whichB = |v − v∗|, and the inverse-power model, in which B = |v − v∗|γb(cos θ) for someexponent γ ∈ R and some complicated function b, which is only known implicitly.Much more details can be found in [42] or in the classical references [20, 21, 22, 41].

Boltzmann’s H theorem identifies D with the entropy production and assertsthat (i) D(f) ≥ 0, (ii) if the collision kernel B is almost everywhere positive, then

(6) D(f) = 0⇐⇒ f = M f .

In other words, the entropy production is nonzero if the distribution function isnot in thermodynamical equilibrium. This theorem is at the heart of most studiesof the hydrodynamical approximation, or the long-time behavior of solutions ofthe Boltzmann equation. This gives strong motivation for establishing quantitativeversions of this theorem. Accordingly, the following question has been studied bymany authors (e.g. [24, 12, 13, 39]):

1With respect to previous papers, I have decided to change my own sign convention for theentropy (unphysical, albeit common in kinetic theory), and accept the idea that it should benondecreasing with time. The influence of Sasha Bobylev in this decision is acknowledged.

2The kernel B is often improperly called the cross-section. Strictly speaking, the cross-sectionwould rather be B(v − v∗, σ)/|v − v∗|.

4 CEDRIC VILLANI

Can one establish a lower bound on the entropy production, in termsof how much the distribution function departs from thermodynamicalequilibrium ?

This question is very representative of recent trends in partial differential equa-tions: in many cases of interest, an entropy principle has been identified, and adetailed study of the entropy production principle provides a sharp insight into theproblem of asymptotic behavior of the equation under consideration [4, 17, 18, 23,35]. In the case of the Boltzmann equation, this problem is interesting not only forthe sake of developing the theory of this particular equation, but also because itpresents a number of interesting mathematical features, some of which are typicalof kinetic equations and some of which are more specific.

In our context, a natural way to measure the departure of f towards thermody-namical equilibrium is by means of the relative H functional,

(7) H(f |M f) = H(f)−H(M f ) =

∫

RN

f logf

Mf.

This is nothing but the relative Kullback information of f with respect to M f .After these preparations, we can state Cercignani’s conjecture: it consists in the

validity of the functional inequality

(8) D(f) ≥ K(f)H(f |M f),

where K(f) would be a positive constant depending on f only via certain a prioriestimates, such as smoothness, decay at infinity or strict positivity.

For the sequel of the discussion, let us introduce the functional norms which weshall use in our estimates. We define the functional spaces L1

s (weigthed L1), L1s logL

(weighted Orlicz space) and Hk (Sobolev space) by the identities

‖f‖L1s

=

∫

RN

f(v)(1 + |v|2)s/2 dv,

‖f‖L1s log L =

∫

RN

f(v)(1 + |v|2)s/2(1 + | log f(v)|

)dv,

and

‖f‖Hk =

∑

|α|≤k

∫

RN

|Dαf(v)|2 dv

12

,

in which α stands for a multi-index of length |α|, and Dαf = ∂α11 . . . ∂αN

N f . Of coursein these definitions we have assumed f ≥ 0.


Cercignani’s conjecture was first formulated with a view to provide simplifiedproofs of convergence to equilibrium for the spatially homogeneous Boltzmann equa-tion; however, later it was understood that it would have important consequenceson the quantitative study of convergence to equilibrium, even in a spatially inho-mogeneous setting. A moment of reflection shows that inequality (8) is essentiallya nonlinear variant of a spectral gap estimate, with all the possible implica-tions that one may imagine. One can make this slightly more precise: a standardlinearization procedure of the Boltzmann equation (with the notation below, writingf = M(1 + εh) and comparing second order terms in ε) transforms inequality (8)into a spectral gap inequality for a linearized Boltzmann operator; therefore thisinequality is stronger than a spectral gap estimate.

At first the constant K(f) in (8) was conjectured to depend only on B, ρ, u andT . Note that, if we define

(9) f(v) =TN/2

ρf(u+

√Tv),

(10) M(v) =e−

|v|2

2

(2π)N/2= M f (v),

then a homogeneity argument yields

(11) H(f |M f) = ρH(f |M), D(f) = ρ2D(f),

where D is the entropy production functional associated with the rescaled collisionkernel

B(v − v∗, σ) = B(√

T (v − v∗), σ).

Therefore it is natural to consider only the case when f lies in the set C1,0,1(RN)

defined by

(12) C1,0,1(RN) =

{f ∈ L1(RN); f ≥ 0; ρ = 1, u = 0, T = 1

}.

Indeed, estimates on this set immediately lead to general estimates by means of (11).As we mentioned above, for a long time only counterexamples appeared in the

subject. The best (negative !) results available are due to Bobylev and Cercig-nani [10]. Roughly speaking, they showed that, for a large class of collision kernels,inequality (8) cannot be true if K(f) is allowed to depend on f only via high-ordermoments, a finite number of Sobolev norms, and “perfect” positivity estimates. Hereis a way to formulate this result precisely.

6 CEDRIC VILLANI

Theorem 1.1 (Bobylev, Cercignani). Let B satisfy∫

SN−1

B(v − v∗, σ) dσ ≤ CB(1 + |v − v∗|γ)

for some γ ∈ [0, 2), CB < +∞. Then there exist sequences (Ms)s∈N, (Sk)k∈N (to beunderstood as moment and smoothness bounds, respectively), and K0, A0 > 0, suchthat

inf

{D(f)

H(f |M); f ∈ C1,0,1 ∩ B

[(Ms), (Sk), A0, K0

]}= 0,

where B[(Ms), (Sk), A0, K0] stands for the space of distribution functions f satisfyingthe bounds

∀s ∈ N, ‖f‖L1s≤Ms

∀k ∈ N, ‖f‖Hk ≤ Sk

∀v ∈ RN , f(v) ≥ K0 e

−A0|v|2 .

Moreover, one can impose that any finite number of the bounds Ms, Sk, K0 and A0

be arbitrarily close to their equilibrium values. More precisely, for all s∗ and k∗, andfor all ε > 0 one can impose

∀s ≤ s∗, ∀k ≤ k∗, Ms ≤ ‖M‖L1s+ε, Sk ≤ ‖M‖Hk+ε K0 ≥ 1−ε, A0 ≤

1

2+ε.

The last part of the theorem rules out even the hope that the conjecture wouldhold true “in the neighborhood of the Maxwellian”. But Theorem 1.1 does notrule out the possibility of Cercignani’s conjecture holding true under more stringentassumptions on f , for instance∫

eα|v|2f(v) dv < +∞.

Barthe [6] has shown to us strong indication that the conjecture might hold true iff , viewed as a reference measure, satisfies a Poincare inequality, a condition whichtypically would require at least

∫eα|v|f(v) dv < +∞. However, in the present state

of the theory of the Boltzmann equation, “polynomial” moment estimates are essen-tially the best we can hope for in a fully nonlinear context (see [9] for an exceptionalcase in which exponential moment estimates are available, however still far fromimplying a Poincare inequality).

At the beginning of the nineties, Carlen and Carvalho [12] were able to proveweaker entropy-entropy production inequalities, in the form

D(f) ≥ Θf

(H(f |M f)

),

where the function Θf would depend on f only via some moment and (mild) smooth-ness estimates, and be strictly increasing from 0. Their results, and even more their


methods, had considerable impact on the field. It was only in 1999 that the resultswere improved [39]: the first polynomial bounds, of the form

D(f) ≥ K(f)H(f |M f)α (α > 1)

were established in a joint paper by Toscani and the author. It was moreover shownthat the exponent α can be taken arbitrarily close to 1 if the collision kernel wasnonvanishing. More precisely, if

(13) B(v − v∗, σ) ≥ KB(1 + |v − v∗|)−β (KB > 0, β > 0),

then for all ε > 0 one had

(14) D(f) ≥ Kε(f)H(f |M f)1+ε.

In this sense Cercignani’s conjecture was shown to be “almost true” for nonvanishingkernels.

Also at the end of the nineties, Carlen, Carvalho and Loss took up the study of anold spectral gap conjecture formulated by Kac. We shall describe it more preciselyin section 6; let us just say here that the main difficulty in Kac’s problem is thatthe dimension n of the phase space grows unbounded, and one wishes to obtainspectral gap estimates which would be independent of the dimension. The conjec-ture, formulated in the fifties, was first solved by Janvresse [28] and independentlyby Maslen [33], no earlier than year 2000. However, the correct formulation forachieving Kac’s goal would rather be an entropic variant of Kac’s problem, namelyfinding “entropy-entropy production constants” Kn which would admit a uniformlower bound, independently of n. However, it is not expected that Kn be universallybounded below, because this would imply a variant of Cercignani’s conjecture, whichis believed to be false as well... In fact Carlen, Carvalho and Loss recently showedthat

K−1n = O(n)

is admissible. Although this has not been explicitly checked, their arguments suggestthat this bound is sharp (what they do prove is that this bound is sharp for aninteresting intermediate entropy inequality; for more information see recent work byCarlen, Lieb and Loss).

From the above presentation we see that many shadow regions remained in thepicture. In this paper we shall enlighten part of them, and present for the firsttime a coherent (although certainly not final) picture of these entropy productioninequalities. Indeed, we shall establish the following three statements:

1) Cercignani’s conjecture holds true when the collision kernel is super-quadratic,in the sense

B(v − v∗, σ) ≥ KB(1 + |v − v∗|2).

8 CEDRIC VILLANI

This seems to be the very first instance in which Cercignani’s conjecture is proven.Moreover, our estimate on the constant K(f) will be surprisingly good. Note thatthis is precisely the limit case which is not covered by Theorem 1.1, so that theassumption of superquadraticity is essentially sharp. We emphasize that this is anonphysical assumption !

As a consequence of this result in the superquadratic case, we shall be able torecover the main results of [39], namely D(f) ≥ Kε(f)H(f |M f)1+ε under assump-tion (13), with a technically simplified proof and improved constants.

2) Cercignani’s conjecture is almost true in all the physically relevant cases. In-deed, we shall extend the results of [39] to cover also situations in which the collisionkernel vanishes for v = v∗ (most typically, the important example of hard spheres,B = |v − v∗|). Since collision kernels coming from physics are bounded below bypositive constants except possibly when |v − v∗| → ∞ or when |v − v∗| → 0, ourresult can be applied in all physically relevant cases. As a trade-off, we shall need ahigher regularity for the distribution function (bounds in all Sobolev spaces).

3) The entropy variant of Kac’s conjecture is also true in the superquadratic case,at least for Kac’s caricature of the Boltzmann equation. At the same time that weestablish this, we shall give a new proof of the result by Carlen, Carvalho and Lossthat, in general, the constant K−1

n can be chosen O(n) as n→∞, in the basic caseof constant collision kernel (see section 6 for details). As a consequence, under thisassumption of superquadratic behavior, we shall be able to realize Kac’s program andconnect the trend to equilibrium of a simple caricature of the Boltzmann equation,with that of a well-chosen many-particle system.

Of these three results, the second is certainly the most technical: it will turn outto be really tricky to handle the problem of small relative velocities without doingany harm to the form of inequality (14). The first and third will be obtained moreeasily, essentially by a re-examination of the method in [39] together with a new useof the Blachman-Stam inequality (see [3, 11] for background on this inequality). Inparticular, we keep from [39] the idea of the auxiliary Ornstein-Uhlenbeck diffusionsemigroup, the use of Landau’s entropy production and the study of symmetries inBoltzmann’s entropy production functional. The Blachman-Stam inequality usedhere is in fact a differential version of the logarithmic Sobolev inequality by Stamand Gross (see [3]). The fact that both our method of proof and logarithmic Sobolev-type inequalities behave well with respect to the dimension will be the key to theproof of the third result.

What are the consequences in terms of convergence to equilibrium ? Of course, ifwe consider f = f(t, v) a solution of the spatially homogeneous Boltzmann equation,satisfying nice bounds, in such a way that (14) holds true, then it is immediate to


prove convergence to equilibrium like O(t−∞), meaning faster than O(t−κ) for allκ. The fact that we need high-order Sobolev regularity for proving (14) in thegeneral case could be seen as a major weakness of our results. This is in fact not thecase: whenever the Cauchy problem can be studied in enough detail, these boundscan be used. In particular, in [34], we shall be able to use this result and recoverconvergence like O(t−∞), even if the solution is not smooth (we shall just need anL2 assumption). The main idea is that the solution can be split into the sum of avery smooth part, and a nonsmooth part which decays exponentially fast.

When one considers the full (spatially inhomogeneous) Boltzmann equation, thenthings are much, much trickier. However, with the help of inequality (14) one canstill prove [25] convergence to equilibrium like O(t−∞), in presence of suitable apriori estimates.

A point which should probably deserve most attention in future studies, is therelationship of (8) to spectral gap estimates. Our methods, which are pretty efficientfor entropy estimates, seem however unable to estimate spectral gaps, in cases whereCercignani’s conjecture does not hold true. For instance, when applied to Kac’sproblem in the case of a constant kernel, it will yield a bound in O(n) for theinverse entropy constant K−1

n , which is optimal — but also a bound in O(n) for theinverse spectral gap, which is not ! The Carlen-Carvalho-Loss strategy is completelydifferent, and manages to treat the spectral gap in a sharp way; maybe a commonframework should be looked for. Further remarks will be formulated in sections 5and 6.

The paper is organized as follows: in each section (except section 5), we state andprove one main theorem. Section 2 establishes the validity of Cercignani’s conjecturefor superquadratic potential. For the sake of completeness, we have included theresome of the arguments in [39] (with a slightly simplified presentation), as well asa simple but crucial auxiliary estimate from [26]. We shall not repeat the proofsalready published in [26, 39], but we thought it would help clarity to re-state thevarious lemmas proven there. Then section 3 shows how to recover the main resultsof [39] from the results in section 2, with a simplified proof and improved constants.Section 4 deals with small relative velocities, and establishes the validity of (14)for all physically realistic cross-sections. In section 5 we present some problemsleft open, and formulate a conjecture about the range of validity of Cercignani’sconjecture. Finally, section 6 discusses the entropy variant of Kac’s problem. Onthis occasion we shall recall a little bit of the history of this topic, and formulateone last open problem.

10 CEDRIC VILLANI

2. Superquadratic collision kernels

In this section, we prove the following theorem.

Theorem 2.1. Let B satisfy

(15) B(v − v∗, σ) ≥ KB(1 + |v − v∗|2)for some KB > 0, and let D be the associated H-dissipation functional. Let f ∈C1,0,1(R

N). Then

(16) D(f) ≥(KB

|SN−1|4(2N + 1)

)(N − T ∗(f))H(f |M),

where

(17) T ∗(f) = maxe∈SN−1

∫

RN

f(v)(v · e)2 dv.

Remark: In words, T ∗f is the “maximum directional temperature” of f . Note that N

is the sum of all directional temperatures. In particular, N −T ∗(f) ≥ (N −1)T∗(f),where T∗(f) stands for the minimum directional temperature of f (defined as in (17)but with “max” replaced by “min”). The strict positivity of T∗(f) is a very weakstrict positivity assumption on f ; it just means that “all directions in R

N” arerepresented in the support of f . The assumption T ∗(f) < N is even weaker, it justmeans that f is not concentrated on a line. In subsequent developments, we shallcrudely bound below (N − T ∗(f))/(4(2N + 1)) by T∗(f)/20.

Corollary 2.2. Let B satisfy (15). Then, for all distribution f on RN there is a

constant K(f), depending only on N , KB, ρ, T and an upper bound for H(f), suchthat

D(f) ≥ K(f)H(f |M).

This corollary follows from Theorem 2.1 by use of (11) and by standard estimateson the positivity of T∗(f), see for instance [26, Proposition 2]. There are many otherways to control the strict positivity of N − T ∗(f), for instance by means of L∞

bounds combined with moment bounds.

Proof of Theorem 2.1. Without loss of generality we assume that B(v − v∗, σ) =KB(1 + |v − v∗|2). As in [39] we shall begin by introducing the adjoint Ornstein-Uhlenbeck (or Fokker-Planck) regularizing semigroup (St)t≥0, defined by the partialdifferential equation

∂f

∂t= ∆f +∇ · (fv) (t ≥ 0, v ∈ R

N).


As shown in [39], under suitable assumptions on f (for instance,∫f | log f |(1 +

|v|s) dv < +∞ for some s > 2), the function D(Stf) is continuous with respect to t,differentiable for t > 0, and goes to 0 as t→∞. In the sequel, we shall assume thatf satisfies these assumptions, and establish (16) for this restricted class of data.Then an approximation argument will imply (16) in the general case. Here is apossible way to implement this approximation argument: without loss of generality,assume that f ∈ L logL. Then replace f(v) by fδ(v) = f(v)e−δ|v|2 , which has fastdecay at infinity for fixed δ. Then note that

(f ′δf

′δ∗ − fδfδ∗) log

f ′δf

′δ∗

fδfδ∗= e−δ(|v|2+|v∗|2)(f ′f ′

∗ − ff∗) logf ′f ′

∗ff∗

.

Therefore, by monotone convergence theorem, D(fδ) −→ D(f) as δ → 0. Of coursea priori fδ /∈ C1,0,1, but one can reduce to this case by using (9) with a rescalingdepending on δ. Details are tedious but easily filled in.

Next, we state four lemmas in a row; each of them is quite easy to check. Thefirst three are reformulations of tools used in [39], and detailed proofs can be foundthere.

Lemma 2.3 (quantitative version of Boltzmann’s angular integration trick).

D(f) ≥ KB|SN−1|4

D(f),

where

D(f) ≡∫

R2N

(ff∗ −

1

|SN−1|

∫

SN−1

f ′f ′∗ dσ

)log

ff∗1

|SN−1|∫

SN−1 f ′f ′∗ dσ

(1+|v−v∗|2) dv dv∗.

This lemma is a direct consequence of Jensen’s inequality and the joint convexityof the function (X, Y ) 7−→ (X − Y )(logX − logY ) on R+ × R+.

Lemma 2.4 (averaging symmetries). The function

G(v, v∗) ≡1

|SN−1|

∫

SN−1

f ′f ′∗ dσ

only depends on

m = v + v∗, e =|v|2 + |v∗|2

2.

This lemma, due to Boltzmann himself, is easy to understand: it suffices to notethat G only depends on the sphere with radius |v − v∗|/2 and center (v + v∗)/2. Aprecise (easy) formulation can be found in [39, Lemma 1].

12 CEDRIC VILLANI

From now on we shall use the notation

F (v, v∗) = ff∗, G(v, v∗) =1

|SN−1|

∫

SN−1

f ′f ′∗ dσ = G(m, e).

Moreover, we shall abuse notations by writing

D(f) = D(F,G).

Rather than bounding directly D from below, we shall do this for D.

Lemma 2.5 (The semigroup is compatible with the symmetries). For brevity, letus use the same notation (St)t≥0 for the adjoint Ornstein-Uhlenbeck semigroup inL1(RN ) and in L1(R2N). Then, for any t ≥ 0,

St(ff∗) = (Stf)(Stf)∗, StG depends only on m and e .

The first part of this lemma is obvious from the explicit representation of St interms of Gaussian functions. As for the second, it is most easily seen by operatingan orthonormal change of coordinates sending (v, v∗) to (v, w) = (v+v∗, v−v∗)/

√2.

Then one just has to notice that St

[φ(v)ψ(w)

]= Stφ(v)Stψ(w), and that St maps

radial functions into radial functions; so if ψ above depends only on |w| = |v−v∗|/√

2,then so does Stψ.

In fact a stronger property holds true, see [39, Proposition 2].In the sequel, we shall use the notation X = (v, v∗) ∈ R

2N . The following lemmais a generalized form of Proposition 5 in [39].

Lemma 2.6 (nonlinear commutator formula). Let (St) be any diffusion semigroupon R

d with generatorL = ∆ + a(X) · ∇+ b(X),

acting on L1(Rd). Then, whenever F and G are smooth densities on Rd,

(18)d

dt

∣∣∣∣t=0

[St

((F −G) log

F

G

)− (StF − StG) log

StF

StG

]

=

∣∣∣∣∇FF− ∇G

G

∣∣∣∣2

(F +G).

The proof is by direct computation. For instance, the object to be computed canbe written as

L

[(F −G) log

F

G

]− (LF − LG) log

F

G− (F −G)

(LF

F− LG

G

).

The terms in a · ∇ cancel out because they represent the action of differentiationoperators, the terms in b disappear because (F − G) log(F/G) is homogeneous of


degree 1. Thus only the terms in ∆ remain (this simplifies a little bit the proofin [39]).

From now on, our treatment departs from that in [39]. Let

ψ(v, v∗) = 1 + |v − v∗|2.Applying Lemma 2.6 with the adjoint Ornstein-Uhlenbeck semigroup (with a(X) =X and b(X) = N), and then integrating with respect to ψ dv dv∗, we find that

− d

dt

∣∣∣∣t=0

D(Stf) =

∫

R2N

ψ(X)

∣∣∣∣∇XF

F− ∇XG

G

∣∣∣∣2

(F +G) dX

−∫

R2N

ψ(X)L

[(F −G) log

F

G

]dX

if f is smooth enough. Here the notation ∇X is used to recall that this is the fullgradient with respect to the 2N scalar variables (v, v∗). If we introduce the adjointoperator

L∗ = ∆X −X · ∇X ,

then we obtain

− d

dt

∣∣∣∣t=0

D(StF, StG) =

∫

R2N

ψ(X)

∣∣∣∣∇XF

F− ∇XG

G

∣∣∣∣2

(F +G) dX

−∫

R2N

L∗ψ(X) (F −G) logF

GdX.

By semigroup property, for all t > 0,

(19) − d

dtD(StF, StG) =

∫

R2N

ψ(X)

∣∣∣∣∇XStF

StF− ∇XStG

StG

∣∣∣∣2

(StF + StG) dX

−∫

R2N

L∗ψ(X) (StF − StG) logStF

StGdX.

To have this formula for t > 0 it is not even necessary to assume that f is smooth,since (St) has a regularizing effect.

By direct computation,

L∗ψ(X) = 4N − 2|v − v∗|2 ≤ 4Nψ(X).

Thus, ∫

R2N

L∗ψ(X) (StF − StG) logStF

StGdX ≤ 4N D(StF, StG).

14 CEDRIC VILLANI

To summarize, we have arrived at

(20)

− d

dtD(StF, StG)+4N D(StF, StG) ≥

∫

R2N

ψ(X)

∣∣∣∣∇XStF

StF− ∇XStG

StG

∣∣∣∣2

(StF+StG) dX.

This is the same as

− d

dt

[e−4NtD(StF, StG)

]≥ e−4Nt

∫

R2N

ψ(X)

∣∣∣∣∇XStF

StF− ∇XStG

StG

∣∣∣∣2

(StF + StG) dX.

Integrating this differential inequality with respect to t, taking into accountD(StF, StG) −→0 as t→∞ and D(F,G) = D(f), we obtain

(21) D(f) ≥∫ +∞

0

e−4Nt

(∫

R2N

ψ(X)

∣∣∣∣∇XStF

StF− ∇XStG

StG

∣∣∣∣2

(StF + StG) dX

)dt.

Complicated as this expression may seem, it will yield the desired bound.Now we borrow another lemma from [39]. In the sequel, we shall denote by Πz⊥

the orthogonal projection onto the hyperplane which is orthogonal to the vectorz ∈ R

N .

Lemma 2.7. Let P be the X-dependent linear operator on R2N ,

P : [A,B] 7−→ Π(v−v∗)⊥ [A− B],

where A and B stand for the components in RNv and R

Nv∗ respectively. To be precise,

P ∈ L∞(R

Nv × R

Nv∗ ; L(R2N , R

N)),

where L(R2N , RN) stands for the set of linear mappings from R

2N to RN . Then, as

soon as G is a smooth density depending only on m and e, one has

P (∇XG) = 0.

As a consequence, whenever G only depends on m and e,∣∣∣∣∇XF

F− ∇XG

G

∣∣∣∣2

≥ 1

‖P‖2L(R2N , RN )

∣∣∣∣P∇XF

F

∣∣∣∣2

=1

2

∣∣∣∣P∇XF

F

∣∣∣∣2

.

The proof is elementary linear algebra, plus the key observation that

∇XG =

[∇mG+ v

∂G

∂e,∇mG+ v∗

∂G

∂e

].

Coming back to our entropy production problem, since StG depends only on mand e (Lemma 2.5) for all t > 0, the action of P will enable one to eliminate this


function from the estimates. On the other hand, the action of P on F (v, v∗) = ff∗is quite nice because of the tensor product structure; in fact, for smooth f ,

∇XF

F=

[∇ff

(v),∇ff

(v∗)

].

Taking into account the first part of Lemma 2.5, we see that, for all t > 0,

∇XStF

StF=

[∇Stf

Stf(v),∇Stf

Stf(v∗)

].

Thus when we apply lemma 2.7 to the right-hand side of (21), we get

D(f) ≥1

2

∫ +∞

0

e−4Nt

(∫

R2N

(1 + |v − v∗|2)∣∣∣∣Π(v−v∗)⊥

[∇Stf

Stf−(∇Stf

Stf

)

∗

]∣∣∣∣2

(StF + StG) dv dv∗

)dt,

in particular(22)

D(f) ≥ 1

2

∫ +∞

0

e−4Nt

(∫

R2N

|v − v∗|2∣∣∣∣Π(v−v∗)⊥

[∇Stf

Stf−(∇Stf

Stf

)

∗

]∣∣∣∣2

(Stf)(Stf)∗ dv dv∗

)dt.

At this point we recall the following lemma, taken from [26, Theorem 1].

Lemma 2.8 (A strong estimate on Landau’s entropy production functional). Let fbe a distribution in C1,0,1(R

N), and let T ∗(f) be the associated maximum directionaltemperature; assume that T ∗(f) < N . Define

(23) DL(f) =1

2

∫

R2N

|v − v∗|2∣∣∣∣Π(v−v∗)⊥

[∇ff−(∇ff

)

∗

]∣∣∣∣2

ff∗ dv dv∗.

Then,

DL(f) ≥ (N − T ∗(f)) I(f |M),

where

I(f |M) =

∫

RN

f

∣∣∣∣∇ logf

M

∣∣∣∣2

is the relative Fisher information of f with respect to M .

The proof of this lemma is performed by first reducing to the case when∫f(v) vi vj dv =

δijTi, which can always be achieved because the matrix∫fv ⊗ v dv is nonnegative

16 CEDRIC VILLANI

symmetric. Then one can expand DL by taking advantage of the quadraticity, andapply the two inequalities

∑

ij

∫(|v|2δij − vivj)

∂if ∂jf

fdv ≥ 0,

∑

i

αi

∫(∂if)2

fdv ≥

∑

i

αi

Ti.

The second one is just a variant of Heisenberg’s inequality. We refer to [26] fordetails.

Since (St) preserves the class C1,0,1, we can use lemma 2.8 to estimate (22), andfind

D(f) ≥∫ +∞

0

e−4Nt(N − T ∗(Stf)) I(Stf |M) dt.

We now apply

Lemma 2.9. For all f ∈ C1,0,1, for all t ≥ 0,

T ∗(Stf) ≤ T ∗(f).

This lemma is easily obtained by noting that whenever e ∈ SN−1, then∫Stf(v) (v · e)2 dv

converges monotonically towards 1, as shown by a simple study of moment behavior.To sum up, at this point we have shown

D(f) ≥ (N − T ∗(f))

∫ +∞

0

e−4NtI(Stf |M) dt.

This is the point where we shall make use of the Blachman-Stam inequality [7,11, 36]. From the representation

Sτg = M1−e−2τ ∗ ge−2τ ,

where we use the notation

gλ(v) = λ−N/2g(v/√λ),

we deduce that

(24) I(Sτg|M) ≤ e−2τ I(g|M) + (1− e−2τ )I(M |M) = e−2τ I(g|M).

This inequality can also be proven by elementary Γ2 theory, since it is at the basis ofthe famous Bakry-Emery argument for proving logarithmic Sobolev inequalities [5].

In the present context, we apply (24) with g = St+2Ntf , and discover that

I(St+2Ntf |M) ≤ e−4NtI(Stf |M).


In particular,

D(f) ≥ (N − T ∗(f))

∫ +∞

0

I(St+2Ntf |M) dt.

Now we change variables in the time-integral, to obtain

(25) D(f) ≥ (N − T ∗(f))

1 + 2N

∫ +∞

0

I(St′f |M) dt′.

The proof of Theorem 2.1 will be completed by the use of the following lemma(see [16] for instance):

Lemma 2.10 (Representation formula for the relative information). Whenever f isa density with finite entropy and finite moments up to order 2, then

H(f |M) =

∫ +∞

0

I(Stf |M) dt.

�

3. Nonvanishing collision kernels

In this section, we shall recover the main result of [39] in a very straightforwardway and in an improved form, as a consequence of Theorem 2.1 plus some easy errorestimate. The main simplification lies in the fact that we do not have to go throughthe complicated error estimate arising in the time-integral along the semigroup St.


(26) B(v − v∗, σ) ≥ KB(1 + |v − v∗|)−β (β ≥ 0).

Let f be a distribution function satisfying

(27) ∀v ∈ RN f(v) ≥ K0e

−A0|v|q0 (K0 > 0, A0 > 0, q0 ≥ 2).

Then, for all ε ∈ (0, 1) there exists a constant Kε(f), depending on ε and dependingon f only through ρ, T , q0, and upper bounds on A0, 1/K0, ‖f‖L1

2+s log L, ‖f‖L12+s+q0

for s = (2 + β)/ε, such that

D(f) ≥ Kε(f)H(f |M)1+ε.

For instance, when ρ = 1, T = 1, the following constant will do:

(28) Kε(f) =KB|SN−1|T∗(f)

40 · 22+βmin

(1

H(f |M),|SN−1|T∗(f)

2s · 40Cε(f)

)ε

,

18 CEDRIC VILLANI

where T∗(f) = mine∈SN−1

∫f(v)(v · e)2 dv as before, and

(29)

Cε(f) = 32 · 2q02 |SN−1| ‖f‖L1

2+s log L‖f‖L12+q0+s

(1 + log

1

K0+ A0

), s =

2 + β

ε.

Remark: This bound is rather crude; we emphasize that there are many possiblevariants of estimate (29), depending on the norms that one is willing to use. Theinterest to have a nice control of the dependence on Cε upon the various boundsabove is demonstrated in [40].

Proof. First we write

B(v − v∗, σ) ≥ KB

(1 +R)2+β(1 + |v − v∗|2)−

KB

(1 +R)2+β(1 + |v − v∗|2)1|v−v∗|≥R.

Accordingly, by Theorem 2.1,

(30) D(f) ≥ KB

(1 +R)2+β

( |SN−1|20

T∗(f)H(f |M)−∫

|v−v∗|≥R

(1 + |v − v∗|2)(f ′f ′∗ − ff∗) log

f ′f ′∗

ff∗dv dv∗ dσ

).

Now we claim that

(31)

∫

|v−v∗|≥R

(1 + |v − v∗|2)(f ′f ′∗ − ff∗) log

f ′f ′∗

ff∗dv dv∗ dσ ≤

Cε(f)

(R/2)s,

where Cε(f) and s are given by (29). If this is true, then Theorem 3.1 followsfrom (30) with the choice

R = 2

(40Cε(f)

|SN−1|T ∗fH(f |M)

)1/s

,

after a few elementary computations. So the proof will be complete when we haveestablished (31). �

Proof of (31). We start with

ff∗ ≥ K20e

−A0(|v|q0+|v∗|q0 ) ≥ K20e

−A0(|v|2+|v∗|2)q0/2

.

Since |v′|2 + |v′∗|2 = |v|2 + |v∗|2, also

f ′f ′∗ ≥ K2

0e−A0(|v|2+|v∗|2)q0/2

.


As a consequence, we have the pointwise inequality

(f ′f ′∗ − ff∗) log

f ′f ′∗

ff∗≤ f ′f ′

∗ log(f ′f ′∗) + ff∗ log(ff∗)

+ (f ′f ′∗ + ff∗)

[log

1

K20

+ A0(|v|2 + |v∗|2)q0/2

].

If we integrate this with respect to (1 + |v − v∗|2) dv dv∗ dσ, and then use the pre-postcollisional change of variables

(v, v∗, σ =

v′ − v′∗|v′ − v′∗|

)←→

(v′, v′∗,

v − v∗|v − v∗|

),

which has unit Jacobian, together with the equality |v − v∗| = |v′ − v′∗|, we obtain∫

|v−v∗|≥R

(1 + |v − v∗|2)(f ′f ′∗ − ff∗) log

f ′f ′∗

ff∗dv dv∗ dσ

≤ 2

∫

|v−v∗|≥R

(1 + |v − v∗|2)ff∗ log(ff∗) dv dv∗ dσ

+ 2

∫

|v−v∗|≥R

(1 + |v − v∗|2)ff∗[log

1

K20

+ A0(|v|2 + |v∗|2)q0/2

]dv dv∗ dσ

By using the inequality (X + Y )q/2 ≤ 2q2−1(Xq/2 + Y q/2) for q ≥ 2, and simple

symmetry tricks, we crudely bound this expression by

(32) 4|SN−1|∫

|v−v∗|≥R

(1 + |v − v∗|2)ff∗ log f dv dv∗

+2q02 |SN−1|

(log

1

K0+ A0

)∫

|v−v∗|≥R

(1+|v−v∗|2)[(1+|v|2)q0/2+(1+|v∗|2)q0/2

]ff∗ dv dv∗.

Now we use the inclusion

(33){|v− v∗| ≥ R

}⊂{|v| ≥ R/2 and |v∗| ≤ |v|

}∪{|v∗| ≥ R/2 and |v| ≤ |v∗|

}.

Accordingly, the first term in (32) can be bounded by

4|SN−1|(∫

|v|≥R/2

(1 + 4|v|2)ff∗ log f dv dv∗ +

∫

|v∗|≥R/2

(1 + 4|v∗|2)ff∗ log f dv dv∗

)

≤ 16|SN−1|(‖f‖L1

∫

|v|≥R/2

(1 + |v|2) f log f dv + ‖f‖L1 log L

∫

|v∗|≥R/2

f∗(1 + |v∗|2) dv∗)

20 CEDRIC VILLANI

≤ 16|SN−1|(‖f‖L1‖f‖L1

2+s log L + ‖f‖L1 log L‖f‖L12+s

(R/2)s

).

The second term in (32) is even easier to bound: indeed, by (33), use of symmetryand application of Fubini’s theorem,∫

|v−v∗|≥R

(1 + |v − v∗|2)[(1 + |v|2)q0/2 + (1 + |v∗|2)q0/2

]ff∗ dv dv∗

≤ 4

∫

|v|≥R/2

(1 + 4|v|2)(1 + |v|2)q0/2 ff∗ dv dv∗

≤ 16‖f‖L1

∫

|v|≥R/2

(1 + |v|2)2+q0

2 f(v) dv

≤ 16‖f‖L1

‖f‖L12+q0+s

(R/2)s.

Putting all these bounds together, one easily concludes to the validity of (31). �

4. General collision kernels

In this section, we establish the following theorem, which generalizes Theorem 3.1to the case when the collision kernel is allowed to vanish for v = v∗ like a power law,under a Sobolev regularity assumption for the distribution function.


(34) B(v − v∗, σ) ≥ KB min(|v − v∗|γ, |v − v∗|−β

), (β, γ ≥ 0),

and let D be the associated H-dissipation functional. Let f be a distribution functionsuch that

(35) ∀v ∈ RN f(v) ≥ K0 e

−A0|v|q0 (K0 > 0, A0 > 0, q0 ≥ 2).

Then, for any ε ∈ (0, 1) there exists a constant Kε(f), depending on f only throughρ, T , q0, and upper bounds on A0, 1/K0, ‖f‖L1

sand ‖f‖Hk , where s = s(ε, q0, β, γ)

and k = k(ε, s, β, γ), such that

D(f) ≥ Kε(f)H(f |M f)1+ε.

For instance, when ρ = 1, T = 1, the following constant will do:

Kε(f) = KB K(N, ε, β, γ)Kε0(f)1+4γ/N

Cε0(f)4γ/Nmin

(1

H(f |M)ε, 1

),

whereε0 = ε/(1 + 4γ/N);


Kε0(f) is defined by (28), in which ε is replaced by ε0;

Cε0(f) =

(1 + log

‖f‖L∞

K0+ A0

)3

‖f‖L12q0+s log L ‖f‖L log L ‖f‖ηL1

q0η +2

log L(1+‖f‖L2)(1+‖f‖2η

Hk),

s = q0

(2

ε0− 1

), k =

2(N + 1)

ε0, η =

ε0

2− ε0.

We have written down this bound so that the reader can have an idea of the ordersof the constants involved in Theorem 4.1, however we insist that this is a quite crudeestimate which admits many possible refinements.

Proof. Without loss of generality, we assume ρ = 1, u = 0, T = 1, so M f = M .Throughout the proof, we shall denote by C and K various numerical constantsdepending only on N , ε, β and γ.

Our starting point is the natural splitting according to whether |v − v∗| is smallor large. Let

B0(v − v∗, σ) = (1 + |v − v∗|)−β,

and let D0 be the associated H-dissipation functional. By assumption,

B(v − v∗, σ) ≥ KBK(β, γ) min(|v − v∗|γ, 1)B0(v − v∗, σ).

Since D is a monotone function of B, we can assume without loss of generality thatB(v − v∗, σ) = min(|v − v∗|γ, 1)B0(v − v∗, σ). So, for all δ ∈ (0, 1), taking intoaccount B0 ≤ 1 we have

B(v − v∗, σ) ≥ δγ[B0(v − v∗, σ)− 1|v−v∗|≤δ

].

Hence,

(36) D(f) ≥ δγ[D0(f)− Dδ(f)],

where we have set

(37) Dδ(f) =1

4

∫

R2N×SN−1

(f ′f ′∗ − ff∗) log

f ′f ′∗

ff∗1|v−v∗|≤δ dv dv∗ dσ.

Let ε0 > 0, to be specified later. This will be an auxiliary small parameter, of thesame order as ε.

From Theorem 3.1 we know that there exists a constant Kε0(f), depending onpositivity, smoothness and moment bounds for f , such that

D0(f) ≥ Kε0(f)H(f |M)1+ε0.

So far there is nothing new: upon replacement of the bounds in [13] by thosein [39], this is the strategy used by Carlen and Carvalho [13] to get rid of small

22 CEDRIC VILLANI

relative velocities. The “standard” way to conclude would be to prove an estimate

like Dδ(f) = O(δκ) for some κ > 0, then optimize in (36) to get a lower bound like

D(f) ≥ KH(f |M)α

for some α > 1.But here, we want to keep the exponent arbitrarily close to 1, so we have to be

more clever. The main difficulty will be to show that not only does Dδ(f) vanishas δ → 0, but also it vanishes up to (almost) order 1 in H(f |M). To state it informulas, we wish to prove

(38) Dδ(f) ≤ Cε0(f)H(f |M)1−ε0δκ.

Note that here ε0 is arbitrarily small, while κ is fixed and does not go to 0 as ε0 → 0.If we plug (38) into (36), we find

D(f) ≥ δγKε0(f)H(f |M)1+ε0

[1− Cε0(f)

Kε0(f)

δκ

H(f |M)2ε0

].

Then the choice

δ =

[Kε0(f)H(f |M)2ε0

2Cε0(f)

]1/κ

leads toD(f) ≥ K ·Kε0(f)1+ γ

κ Cε0(f)−γκ H(f |M)1+ε0(1+ 2γ

κ ),

which is what we are looking for (with ε = ε0(1 + (2γ/κ))). So the proof will becomplete when we have shown the theorem below. �

Theorem 4.2. Let f ∈ C1,0,1. For any ε0 ∈ (0, 1) there exists a finite constantCε0(f), depending on f onl via estimates of positivity, decay and smoothness, suchthat

∀δ ∈ (0, 1), Dδ(f) ≤ Cε0(f)H(f |M)1−ε0δN/4.

Proof of Theorem 4.2. The proof is not so long but quite tricky. We divide it intofive steps.

Step 1: Introduction of the Maxwellian distribution M by force. We shall use thefollowing elementary inequality:

Lemma 4.3. Let X, Y , Z be three positive real numbers. Then

(X − Z) logX

Z≤ Cmax

(1, log

X

Z, log

Z

X

) [(X − Y ) log

X

Y+ (Y − Z) log

Y

Z

],

where C is a numerical, computable constant.


Before we prove this lemma, let us explain how we shall use it. Choose

X = f ′f ′∗, Z = ff∗, Y = MM∗ = M ′M ′

∗.

From the assumption on f we have

‖f‖L∞ ≡ C0 ≥ f ≥ K0e−A0|v|q0 ,

soC2

0 ≥ ff∗ ≥ K20e

−A0(|v|q0+|v∗|q0).

Just as in the proof of Theorem 3.1,

C20 ≥ f ′f ′

∗ ≥ K20e

−A0(|v|2+|v∗|2)q0/2

.

Thus

max

(1, log

X

Z, log

Z

X

)≤ 2

(1 + log

C0

K0+ A0(|v|2 + |v∗|2)q0/2

).

Applying Lemma 4.3, we deduce

(39) (f ′f ′∗ − ff∗) log

f ′f ′∗

ff∗≤ C

(1 + log

C0

K0+ A0(|v|2 + |v∗|2)q0/2

)

[(f ′f ′

∗ −M ′M ′∗) log

f ′f ′∗

M ′M ′∗

+ (ff∗ −MM∗) logff∗MM∗

]

(40) = C

(1 + log

C0

K0+ A0(|v′|2 + |v′∗|2)q0/2

)(f ′f ′

∗ −M ′M ′∗) log

f ′f ′∗

M ′M ′∗

+ C

(1 + log

C0

K0+ A0(|v|2 + |v∗|2)q0/2

)(ff∗ −MM∗) log

ff∗MM∗

,

by energy conservation again.We next multiply (40) by 1|v−v∗|≤δ and integrate with respect to dσ dv dv∗. Using

the pre-postcollisional change of variables and the equality |v− v∗| = |v′− v′∗| again,we end up with

Dδ(f) ≤ C

(1 + log

C0

K0+ A0

)∫

|v−v∗|≤δ

(ff∗ −MM∗) logff∗MM∗

(1 + |v|2 + |v∗|2)q0/2 dv dv∗ dσ

(41)

= C|SN−1|(

1 + logC0

K0+ A0

)∫

|v−v∗|≤δ


(1 + |v|2 + |v∗|2)q0/2 dv dv∗,

(42)

where C is a numerical constant. This is the end of Step 1.Before going further, we shall give the proof of Lemma 4.3.

24 CEDRIC VILLANI

Proof of Lemma 4.3. By symmetry and homogeneity, we may assume without lossof generality that X ≥ Z and Y = 1. Then what we wish to prove is

(X − Z) logX

Z≤ C max

(1, log

X

Z

)[(X − 1) logX + (Z − 1) logZ].

We distinguish six cases.

Case 1: 12≤ X ≤ 4, 1

10≤ Z ≤ 4. Then

(X − Z) logX

Z≤ C(X − Z)2 ≤ C[(X − 1)2 + (Z − 1)2]

≤ C[(X − 1) logX + (Z − 1) logZ].

Case 2: 12≤ X ≤ 4, Z < 1

10. Then

(X − Z) logX

Z≤ C log

1

Z≤ C(Z − 1) logZ.

Case 3: X ≤ 12, X ≥ Z ≥ X/2. Then

(X − Z) logX

Z≤ C ≤ C(X − 1) logX.

Case 4: X ≤ 12, Z < X/2. Then

(X − Z) logX

Z≤ X log

X

Z= X logX +X log

1

Z≤ C + C log

1

Z≤ C(Z − 1) logZ.

Case 5: X ≥ 4, X/2 ≤ Z ≤ X. Then

(X − Z) logX

Z≤ C(X − Z) ≤ CX ≤ C(X − 1) logX.

Case 6: X ≥ 4, Z ≤ X/2. Then

(X − Z) logX

Z≤ X log

X

Z≤ C log

X

Z

((X − 1) logX

).

This concludes the proof of the lemma. �


Step 2: Tail estimates. Let R > 0, and

ER(f) =

∫

|v|2+|v∗|2≥R2/2


(1 + |v|2 + |v∗|2)q0/2 dv dv∗.

By a proof strictly analogous to the one in section 3, for all s > 0 one can establishthe estimate

(43) ER(f) ≤ 2q02

+1

(‖f‖L1

‖f‖L1q0+s log L

(R/2)s+‖f‖L1

q0+s

(R/2)s‖f‖L log L

+‖M‖L1

‖M‖L1q0+s log L

(R/2)s+‖M‖L1

q0+s

(R/2)s‖M‖L log L

)

+ 2q0

(log

1

K ′20

+ A′0

)(‖f‖L12q0+s

(R/2)s‖f‖L1 +

‖M‖L12q0+s

(R/2)s‖M‖L1

),

where K ′0 = inf(K0, (2π)−N/2), A′

0 = max(A0, 1/2). We abbreviate this by

ER(f) ≤ Cs(f)

Rs,

where Cs(f) = C(1 + logK−10 + A0)‖f‖L1

2q0+s log L‖f‖L log L.

If we now plug this estimate into (41), we find

Dδ(f) ≤

C

(1 + log

1

K0

+ A0

){(1 +

R2

2

)q0/2 (∫

|v−v∗|≤δ


dv dv∗

)+Cs(f)

Rs

}.

A proper choice of R shows that

Dδ(f) ≤ C

(1 + log

1

K0+ A0

)Cs(f) max

{(∫

|v−v∗|≤δ


dv dv∗

) ss+q0

,

∫

|v−v∗|≤δ


dv dv∗

}

We shall chose s in such a way that

s

s+ q0= 1− ε0

2.

26 CEDRIC VILLANI

So all we have to prove is the following: for all η ∈ (0, 1), there exists C ′η(f), only

depending on suitable estimates on f , such that

(44)

∫

|v−v∗|≤δ


dv dv∗ ≤ C ′η(f)δ

N2 H(f |M)1−η.

Then the proof of Theorem 4.2 will be complete upon choosing η = ε0/(2− ε0), sothat (1− η)(1− ε0/2) = 1− ε0.

Step 3: Localization to physical space. This step is crucial. The error termappearing in the left-hand side of (44) has the form of an integral over a thin stripin R

2N ; and we wish to replace it by integrals over small balls in RN .

For this we proceed as follows. First, by symmetry,∫

|v−v∗|≤δ


dv dv∗ = 2

∫

|v−v∗|≤δ

(ff∗ −MM∗) logf

Mdv dv∗.

Next, we rewrite the integrand as(45)

(ff∗−MM∗) logf

M= f∗(f−M) log

f

M−(f∗−M∗)

(M log

M

f−M + f

)+(f∗−M∗)(f−M).

Note that in this decomposition we have carefully isolated the dominant terms inthe form of nonnegative expressions ! This will be crucial to avoid destroying thestructure of the integrand. A less careful decomposition would lead to the appear-ance of terms like f | log f |, at the level of (47) for instance, and then we would belost for the last steps of the proof.

As a consequence, we obtain

(46)

∫

|v−v∗|≤δ


dv dv∗ =

2

∫

|v−v∗|≤δ

f∗(f−M) logf

Mdv dv∗−2

∫

|v−v∗|≤δ

(f∗−M∗)

(M log

M

f−M + f

)dv dv∗

+ 2

∫

|v−v∗|≤δ

(f∗ −M∗)(f −M) dv dv∗.

We shall estimate separately the three terms appearing in (46).First of all,

∫

R2N

f∗1|v−v∗|≤δ(f−M) logf

Mdv dv∗ ≤ sup

v∈RN

(∫

RN

f∗1|v−v∗|≤δ dv∗

)∫

RN

(f−M) logf

Mdv.


By Cauchy-Schwarz inequality, for any v ∈ RN ,

∫

RN

f∗1|v−v∗|≤δ dv∗ ≤ ‖f‖L2

√∫

RN

1|v−v∗|≤δ dv∗ ≤ |BN | 12 ‖f‖L2δN/2,

where |BN | stands for the volume of the unit ball in RN . Thus,

(47)

∫

R2N

f∗1|v−v∗|≤δ(f −M) logf

Mdv dv∗ ≤ |BN | 12 ‖f‖L2δN/2

∫

RN

(f −M) logf

M.

We next treat similarly the second term in (46):∣∣∣∣∫

|v−v∗|≤δ

(f∗ −M∗)

(M log

M

f−M + f

)∣∣∣∣

≤ supv∈RN

(∫|f∗ −M∗|1|v−v∗|≤δ dv∗

)∫

RN

(M log

M

f−M + f

)dv

≤ |BN | 12 (‖f‖L2 + ‖M‖L2)δN/2

∫ (M log

M

f−M + f

)dv.

As for the last term in (46), we have∣∣∣∣∫

|v−v∗|≤δ

(f∗ −M∗)(f −M) dv dv∗

∣∣∣∣ ≤ supv∈RN

(∫

RN

1|v−v∗|≤δ|f∗ −M∗| dv∗)∫

|f −M | dv

≤ |BN | 12 δN/2‖f −M‖L2‖f −M‖L1 .

Putting all previous estimates together, we end up with

(48)

∫

|v−v∗|≤δ


dv dv∗ ≤ 2|BN | 12 (‖f‖L2 + ‖M‖L2 + 1)δN/2

×(∫

(f −M) logf

M+

∫ (M log

M

f−M + f

)+ ‖f −M‖L2‖f −M‖L1

).

Since∫f =

∫M , this can be rewritten in terms of relative informations:

(49)

∫

|v−v∗|≤δ


dv dv∗ ≤ 2|BN | 12 (‖f‖L2 + ‖M‖L2 + 1)δN/2

×[H(f |M) +H(M |f) + ‖f −M‖L2‖f −M‖L1

].

This is the end of Step 3. To conclude the proof of (44), it will suffice to show that

(50) ‖f −M‖L2‖f −M‖L1 ≤ C ′′η (f)H(f |M)1−η.

(51) H(M |f) ≤ C ′′η (f)H(f |M)1−η,

28 CEDRIC VILLANI

These inequalities will be proven in Steps 4 and 5 respectively.

Step 4: Interpolations. In this step we prove (50). The idea is to reduce to L1

norms in order to apply the Csiszar-Kullback-Pinsker inequality,

(52) ‖f −M‖L1 ≤√

2H(f |M).

So we use the following interpolation lemma:

Lemma 4.4. Let u be a smooth function on RN ; then, for all θ ∈ (0, 1) there is a

numeric constant Cθ = C(θ,N) such that

(53) ‖u‖L2(RN ) ≤ Cθ‖u‖1−θL1(RN )

‖u‖θHk(RN ), k =N + 1

2θ.

Remark: Any k > N(1− θ)/(2θ) would do.

Once this lemma is proven, inequality (50) will follow by choosing θ = 2η andapplying (52).

Proof of Lemma 4.4. Let u stand for the Fourier transform of u:

u(ξ) =

∫

RN

e−ix·ξu(x) dx.

Then,

‖u‖2L2 =1

(2π)N‖u‖2L2 =

1

(2π)N

∫

RN

|u(ξ)|2(1 + |ξ|)(N+1)(1−θ)

(1 + |ξ|)(N+1)(1−θ) dξ

≤ 1

(2π)N

(∫ |u(ξ)|2(1 + |ξ|)N+1

dξ

)1−θ (∫

RN

|u(ξ)|2(1 + |ξ|)(N+1)(1−θ)

θ dξ

)θ

thanks to Holder’s inequality. Then,

‖u‖2L2 ≤ C(N)

(supξ∈RN

|u(ξ)|2)1−θ (∫

RN

dξ

(1 + |ξ|)N+1

)1−θ (∫

RN

|u(ξ)|2(1 + |ξ|)N+1θ dξ

)θ

≤ C(θ,N) ‖u‖2(1−θ)

L1 ‖u‖2θ

H(N+1)

2θ

.

In the last inequality we have used the well-known representation of Sobolev normsby Fourier transform. This concludes the proof. �

Step 5: From H(M |f) to H(f |M). In this last step, we prove inequality (51).For this we shall again use positivity and moment estimates.


First we write

H(M |f) =

∫M log

M

f=

∫ (f

M− log

f

M− 1

)M dv.

In this way we shall be able to take advantage of the quadratic behavior of thefunction X − logX − 1.

Lemma 4.5. There exists a numeric constant CL such that for all X ∈ R,

X − logX − 1 ≤ CL max

(1, log

1

X

)(X logX −X − 1).

Proof. The proof is immediate if one separates the three casesX ≤ 1/2, 1/2 ≤ X ≤ 2and X ≥ 2. �

From this lemma we deduce

H(M |f) ≤ CL

∫max

(1, log

M

f

)(f

Mlog

f

M− f

M+ 1

)M dv

= CL

∫max

(1, log

M

f

)(f log

f

M− f +M

)dv

≤ CL

[∫ (f log

f

M− f +M

)dv

]1−η[∫

max

(1, log

M

f

)1/η (f log

f

M− f +M

)dv

]η

.

In order to conclude the proof of (44), we just have to bound the integral on theright-hand side. For this we note that

(54) max

(1, log

M

f

)≤ max

(1, log

1

K0

)+ A0|v|q0,

so a rough estimate leads to

∫max

(1, log

M

f

)1/η (f log

f

M− f +M

)dv

≤ C(N)

[max

(1, log

1

K0

)+ A0

]1/η (‖f‖L1

q0η

log L + ‖f‖L1q0η +2

+ ‖M‖L1q0η

).

This concludes the proof of Theorem 4.2. �

Remark: Handling rough distributions. One can wonder what can still beproven if one has to deal with density functions which do not possess the niceproperties of decay, smoothness and positivity which we used above. This was thesituation in the original Carlen-Carvalho results [12, 13]. If one is ready to accept

30 CEDRIC VILLANI

polynomial lower bounds of the form D(f) ≥ H(f |M f)αf , where the exponent αwould not necessarily be close to 1, then the assumptions on f can be considerablyrelaxed:

• It is very easy to relax the moment assumption. This can be achieved by thefollowing standard way. Let χ(v) be a radially symmetric smooth cut-off function,which will be fixed once for all in such a way that χ ≥ 0, χ(v) = 1 for |v| ≤ 1,χ(v) = 1/|v|µ for |v| ≥ 2, where µ will be chosen later on. We define

fR(v) = f(v)χ( vR

);

then we apply our main theorem to fR. Using the crude moment bound andthe smoothness, it is possible to show that H(fR|MfR) is not much smaller thanH(f |M f), and that D(fR) is not much bigger than D(f); error terms for this aregiven as a function of R. Then, an optimization in R yields the result.

• If the kernel does not vanish, then it is also very easy to relax the smoothness andthe lower bound assumption. Indeed, by crude moment bounds, we may assumethat B ≥ 1. Then, if Pt stands for the heat semigroup, we have D(Ptf) ≤ D(f)by convexity of f ⊗ f 7−→ D(f). But Ptf is smooth and bounded from below; sothe only thing to know is that H(Ptf |MPtf) is not much smaller than H(f |M).This only needs an ad hoc assumption on the modulus of continuity of the functionH(Ptf) in terms of t. For instance, if f has a finite Fisher information,

I(f) =

∫ |∇f |2f

< +∞,

then one has the bound

H(f)−H(Ptf) =

∫ t

0

I(Pτf) dτ ≤ I(f)t,

because I(Ptf) is a nonincreasing function of t (this property is well-known to spe-cialists; see [45] for a short proof).

• Even if the kernel vanishes, then it is possible to require less smoothness onthe distribution. An example of such computation is in [12], which uses Lipschitzbounds.

5. Further developments and open problems

A natural question to ask for now, is whether Cercignani’s conjecture holds true forother collision kernels than the quadratic ones. As we saw, the answer is negativeif∫B(v − v∗, σ) dσ = O(1 + |v − v∗|γ) for some γ < 2. However, a physically

interesting case is not covered by these assumptions, namely the one in which B


presents a nonintegrable angular singularity for small deviation angles. In this case,the Boltzmann operator resembles a fractional derivation operator (see [2]) and onecould wonder whether this has a chance to help. For instance, it is shown in [2]that for some kernels of this kind, finiteness of D(f) implies finitess of a well-chosenfractional Sobolev norms of

√f .

At present, we do not know how to answer this question, which seems extremelyintricate. However, we can formulate a guess as follows.

The links between Boltzmann’s and Landau’s entropy production functionals havebeen studied in detail (see for instance [1, 43]). In particular, the following is known.Let Bε(v − v∗, σ) be a family of Boltzmann collision kernels taking the special form

Bε(v − v∗, σ) = Φ(|v − v∗|) bε(cos θ), cos θ =

⟨v − v∗|v − v∗|

, σ

⟩,

where the angular dependence bε satisfies∫ π

0

bε(cos θ)(1− cos θ) sinN−2 θ dθ −−→ε→0

µ > 0,

∀θ0 > 0, supθ≥θ0

bε(cos θ) −−→ε→0

0.

Then, for any fixed smooth density f , one has

DBε(f) −−→ε→0

DL(f),

where DBε stands for Boltzmann’s entropy production functional with kernel Bε,and DL stands for Landau’s entropy production functional, in the form

(55) DL(f) = 2

∫

R2N

Ψ(|v − v∗|)∥∥∥Π(v−v∗)⊥

[∇√f − (∇

√f)∗]∥∥∥

2

,

Ψ(|z|) = CN µΦ(|z|)|z|2.In particular, if one considers a family of collision kernels of the form

(56) B(v − v∗, σ) = (1 + |v − v∗|γ) b(cos θ),

where b(cos θ) is a positive kernel satisfying∫ 2π

0

b(cos θ)(1− cos θ) sinN−2 θ dθ = 1

and presenting a singularity of order 1 + ν at θ = 0, then the Landau entropyproduction (23) appears as the limit case ν = 2, γ = 0. More generally, for a givenpower γ, the limit case ν = 2 would correspond to (55) with Ψ behaving like apower γ + 2. In fact, most of the time the quantity γ + ν has a lot of influence on

32 CEDRIC VILLANI

the qualitative properties of the Boltzmann operator when grazing collisions havean important role [2].

Now, the main result of the present paper states that Cercignani’s conjecture holdstrue in the case γ = 2, ν = 0, while the main result of [26] states that this conjecturealso holds true in the limit case γ = 0, ν = 2. In view of the above considerations,it is very tempting to conjecture that we have just identified the extremal points ofa family of inequalities governed by the quantity γ + ν. Thus we are led to the

Conjecture. Let B(v − v∗, σ) be a collision kernel of the form (56), where b hasan angular singularity of order 1 + ν at θ = 0 (ν ≥ 0; by convention ν = 0 if b isintegrable). Then, Cercignani’s conjecture holds true if and only if γ + ν ≥ 2. Onthe other hand, the linearized Boltzmann collision operator admits a spectral gap ifand only if γ ≥ 0.

Would this conjecture be true, it would mean that Cercignani’s conjecture isalways false in realistic cases; indeed, classical physics yields exponents γ and νsatisfying γ + ν ≤ 1. It would also lead to a remarkable conclusion, namely thatthe physical quantities which matter for an entropy-entropy production inequalityto hold, differ from those which matter for a spectral gap inequality to hold.

As we mentioned above, another problem in which questions naturally arise is anentropic version of Kac’s famous spectral gap problem for many particles. As weshall see in the next section, our estimates can be adapted for such a purpose, andlead to the first accomplishment of Kac’s program on a simplified artificial model ofBoltzmann equation.

6. The entropy variant of Kac’s problem

At the beginning of the fifties, Kac [30] had the idea to attack the problem of trendto equilibrium for the Boltzmann equation, by a study of a “master equation”, i.e.a linear equation for the density function of a many-particles system. The mainapparent advantage of this approach was to trade the complexity of the nonlinearBoltzmann equation for the simplicity of a linear equation, however on a many-particle phase space. It seems natural to use the tools of spectral theory to studythe asymptotic behavior of the solution of this linear equation, and this might haveled to a natural road towards convergence theorems for the nonlinear Boltzmannequation, for which no way of attack could be seen. The problem set by Kac, andwhich he was unable to achieve at the time, was to prove that a certain family oflinear operators Ln, defined on larger and larger phase spaces (the phase spaces forn particles), admitted a uniform lower bound on the size of their spectral gap λn.


Let us give here the basic example, commonly called Kac’s caricature of aMaxwell gas. In this model, the particles are modelled by n one-dimensional ve-locities, evolving stochastically as follows. Attached to the system is a Poisson clockwith rate n; whenever it rings, two distinct particles i and j are chosen randomly,and they change states from the old velocities (vi, vj) to the new values

(v′i, v′j) = Rθ(vi, vj),

where Rθ is the counterclockwise rotation of angle θ in the plane, and θ is chosenrandomly (uniformly) in [0, 2π]. Since these “collisions” preserve kinetic energy, thenatural phase space is the sphere

√nSn−1 (so that each particle has energy 1 on the

average). The linear operator describing this evolution is

(57) Lnf(n) =

n( n

2

)∑

i<j

2π

−∫

0

[f (n) ◦Rijθ − f (n)] dθ,

where Rijθ stands for the counterclockwise rotation of angle θ in the (i, j) plane, and

−∫

stands for the normalized integral. Since the clock typically rings n times over aunit time interval, each particle typically collides once, and in the limit as n →∞,under a chaos assumption, the one-particle marginal can be shown to satisfy a simpleanalogue of Boltzmann’s equation,

(58)∂f

∂t=

∫

R

2π

−∫

0

f(v′)f(v′∗) dθ dv∗

− f, t ≥ 0, v ∈ R,

where (v′, v′∗) = Rθ(v, v∗).As a by-product, these studies led Kac to introduce the notion of propagation

of chaos, and one of the first mathematical treatments of mean-field limit for acontinuous phase space. There is hardly any doubt that these by-products were byfar more important than Kac’s original goal. The study of chaos and mean-fieldlimits underwent important progress in the following decades [37, 38], while thespectral gap study of the master equations was stalled. Kac implicitly conjecturedthat the spectral gap λn of (57) was uniformly bounded below as n→∞. Recently,Diaconis and Saloff-Coste [27], using a group-theoretical approach, could not obtainbetter than λ−1

n = O(n2) for the spectral gap λn of the operator (57). It was onlytwo years ago that Janvresse [28] finally solved Kac’s conjecture; for this she usedthe so-called Yau entropy method. Shortly after, Carlen, Carvalho and Loss [14]dramatically improved this result by computing the exact value of the spectral gap,introducing a deep and general induction argument, by which they also could treatmore general versions of Kac master equations, in particular the one corresponding to

34 CEDRIC VILLANI

Boltzmann’s equation for Maxwell molecules. Independently, Maslen [33] managedto determine the whole spectrum of the Kac operator by a completely differentmethod.

However, it was soon realized that these results would not help solving Kac’soriginal problem, namely convergence to equilibrium for a nonlinear equation set ona one-particle phase space. What is the problem ? Let us still consider the caseof Kac’s caricature: the phase space is

√nSn−1, and spectral gap tells about the

speed of convergence in L2(√nSn−1). But the norm of the n-particle distribution

function f (n) in L2(√nSn−1) is very hard to relate to relevant quantities about the

one-particle marginal f , even when f is very close to be in tensor product form.Roughly speaking, it increases exponentially with n, and cannot be asymptoticallyexpressed in terms of some simple norm of f , see [30]. Even if nobody has everchecked this explicitly, it seems that Kac’s approach rather leads to estimates aboutthe speed of convergence for a linearized Boltzmann-type equation.

Now, if one still wants to achieve Kac’s program and relate the speed of con-vergence for the nonlinear Boltzmann equation, to the speed of convergence for amany-particles system, the natural thing to do is to look at this problem in termsof entropy and entropy production. Indeed, entropy behaves very well for problemswith arbitrarily large phase space, a property which has been used again and againin the context of hydrodynamic limits of particle systems [31]. The entropy versionof Kac’s problem now becomes:

Find an asymptotically sharp lower bound, as n→∞, on

Kn ≡ inff(n)

D(f (n))

H(f (n)),

where f (n) describes the set of probability densities on√nSn−1 with finite entropy,

D(f (n)) stands for the entropy production of the system of n particles under consid-eration, and H(f (n)) for the negative of its entropy.

In [29], Janvresse gives some hint of why Yau’s method seems to fail on thisproblem. It was shown in the author’s master thesis [44] that K−1

n = O(n). Thisestimate was first believed by the author to be suboptimal; but recent work byCarlen, Lieb and Loss [15], using a completely different method, suggests that it isoptimal. The fact that K−1

n is not uniformly bounded prevents from finding anyrelevant conclusion at the level of the limit n →∞; this is consistent with the factthat Cercignani’s conjecture is not believed to hold true for equation (58).

Now, what we prove by an adaptation of the proof of Theorem 2.1, is the followingmore general result. In the case γ = 2, it includes the first Kac-type model for whichthe entropy production problem can be solved in a satisfactory way.


Theorem 6.1. Let γ ∈ [0, 2]. Consider the following Kac-type master equation:

(59)∂f (n)

∂t= Lnf

(n) ≡ n( n

2

)∑

i<j

2π

−∫

0

[1 + (v2

i + v2j )

γ/2](f (n) ◦Rij

θ − f (n))dθ,

t ≥ 0, v ∈√nSn−1,

where the unknown f (n) is a probability density on the sphere√nSn−1. Define

H(f (n)) =

∫√

nSn−1

f (n) log f (n),

D(f (n)) =1

2

∫√

nSn−1

Ln(f (n))(log f (n) ◦Rijθ − log f (n)).

Then

D(f (n)) ≥ nγ/2

(γ + 1)n− 1H(f (n)) ≥ H(f (n))

(γ + 1)n1− γ2

.

In particular, for γ = 2, one has

D(f (n)) ≥ H(f (n))

3independently of n.

Remarks:1. Note that v2

i +v2j is typically of order 1, so our modified Kac-type linear operator

is of the same order of magnitude than the one appearing in Kac’s caricature of aMaxwell gas. It describes the same system as Kac’s model, except that now eachpair of particles (i, j) has its own Poisson clock, ringing with rate (v2

i + v2j )

γ if thetwo particles are in states vi, vj respectively. The collision between particles i andj is performed only when the corresponding clock has rung.

2. In the case γ = 2, we deduce from this study that

H(f(n)t ) ≤ H(f

(n)0 )e−t/5,

where f(n)t stands for the n-particle probability density at time t. Assume that f

(n)0

is a “strongly” chaotic sequence of initial data, for instance the restriction of a tensorproduct probability measure to the sphere

√nSn−1. Then3 one has

H(f0|M) = limn→∞

H(f(n)0 )

n.

3Added note: The remark is naive but essentially true; see the notes at the end of this section.

36 CEDRIC VILLANI

On the other hand, from a propagation of chaos argument and the properties of Hone can show that for any t ≥ 0,

H(ft|M) ≤ lim infn→∞

H(f(n)t )

n,

where ft stands for the solution of (58) at time t. Thus the convergence to equi-librium for the limit equation can be deduced from the estimates on the n-particlesystem. Note that in the present case, it could also have been established directly.

3. By a standard linearization procedure, this bound on the entropy productionimplies an estimate of the spectral gap, with the same constants. In the case γ = 0,the estimate obtained in this way already improves on the Diaconis-Saloff-Costeresult [27], by a factor n; but does not (cannot ?) catch the optimal results of [28]or [14].

4. We conjecture that the estimate K−1n = O(n(1−γ/2)) is optimal. This could

possibly be checked by a reexamination of the methods by Carlen, Carvalho andLoss [14], and by Carlen, Lieb and Loss [15].

5. Here is the conclusion that can be drawn from this study, when compared withthe observations by Carlen, Lieb and Loss [15]: in the entropy production problemfor n particles with γ = 0, very low entropy production rates can be achieved byconsidering particular configurations which are very “unfair”, in the sense that just afew (a small fraction of N , however nonvanishing in the limit N →∞) fast particlescarry an important fraction of the energy. If one enhances the rate of interaction ofthese fast particles (by introducing v2

i + v2j in the collision kernel), then this effect

can be compensated for. It should be noted that the spectral gap inequality does notworry about the existence of these “unfair” energy distributions. As in the previoussection, we see that entropy production estimates are more sensitive to the dynamicsof the particle system.

Proof of Theorem 6.1. It is in the same spirit as that of Theorem 2.1. The maindifference is that the use of the Blachman-Stam inequality will be replaced by theuse of a classical estimate coming from the theory of logarithmic Sobolev inequalitieson manifolds.

To begin with, we use the inequality 1 +Xα ≥ (1 +X)α (X ≥ 0, 0 ≤ α ≤ 1), toreplace the kernel 1 + (v2

i + v2j )

γ/2 by its lower bound (1 + v2i + v2

j )γ/2. The interest

of this will appear later.


To simplify notations we rescale everything to work on the unit sphere Sn−1. Letσn be the uniform probability on Sn−1. The problem becomes: bounding below

D(f (n)) =n

2( n

2

)∑

i<j

1

2π

∫ 2π

0

∫

Sn−1

[1+n(v2

i +v2j )]γ/2

[f (n)◦Rijθ −f (n)] log

f (n) ◦Rijθ

f (n)dσn dθ

in terms of

H(f (n)) =

∫

Sn−1

f (n) log f (n) dσn,

whenever∫

Sn−1 f(n) dσn = 1. To alleviate notations we shall drop the subscript (n)

for the probability density.By Jensen’s inequality,

D(f) ≥ D(f),

where

D(f) =n

2( n

2

)∑

i<j

∫

Sn−1

[1 + n(v2

i + v2j )]γ/2

(f − f ij) logf

f ijdσn,

and

f ij(v) =1

2π

∫ 2π

0

f(Rijθ (v)) dθ.

Let ∆S be the Laplace-Beltrami operator on the sphere Sn−1. It can be defined as

∆S =∑

i<j

(Dij)2 =1

2

∑

ij

(Dij)2,

where

Dijf = vi∂f

∂vj− vj

∂f

∂vi=

d

dθ

∣∣∣∣θ=0

f ◦Rijθ .

An easy computation shows that∫

Sn−1

f∆Sf = −∫

Sn−1

|∇f |2 dσn,

where ∇f stands for the tangential gradient on Sn−1.Let (St)t≥0 be the heat semigroup on the sphere, generated by the Laplace-

Beltrami operator. A computation similar to the one in the proof of Theorem 2.1

38 CEDRIC VILLANI

shows that

(60) − d

dtD(Stf) =

n

2( n

2

)∑

i<j

∫

Sn−1

φij(Stf + Stfij)|∇ logStf −∇ logStf

ij|2 dσn

− n

2( n

2

)∑

i<j

∫

Sn−1

∆Sφij(Stf − Stfij) log

Stf

Stf ijdσn,

where φij(v) = [1+n(v2i +v2

j )]γ/2. Here we have used the fact that (Stf)ij = St(f

ij).Next, by direct computation,

∆S(v2i + v2

j ) = 4− 2n(v2i + v2

j ) ≤ 4,

in particular, from the formula ∆aγ/2 ≤ (γ/2)(∆a)/a1−γ/2, we deduce

∆Sφij(v) ≤2nγ

[1 + n(v2i + v2

j )]1−γ/2

≤ 2nγφij(v).

Let Pij stand for the orthogonal projection onto the (i, j) plane: clearly,

(Stf + Stfij)|∇ logStf −∇ logStf

ij|2 ≥ Stf |Pij∇ log Stf − Pij∇ logStfij|2.

Now, the crucial symmetry argument on which the proof relies is that Pij∇ logStfij

is always colinear to (vi, vj). Let Tij ∈ L∞(Sn−1;L(R2,R)) be the v-dependent linearoperator with unit norm defined by

Tij : [Xi, Xj] 7−→Xjvi −Xivj√

v2i + v2

j

.

Then TijPij∇ logStfij = 0, and thus

|Pij∇ logStf − Pij∇ log Stfij|2 ≥ (TijPij∇ logStf)2

=(Dij logStf)2

v2i + v2

j

.

Since v ∈ Sn−1, we have

φij

v2i + v2

j

≥ nγ/2

(v2i + v2

j )1− γ

2

≥ nγ/2.


All in all, we arrive at

− d

dtD(Stf) ≥ n

2( n

2

)∑

i<j

∫

Sn−1

nγ/2Stf(Dij logStf)2 dσn

− n

2( n

2

)∑

i<j

∫

Sn−1

2γnφij(Stf − Stfij) log

Stf

Stf ijdσn,

which implies

− d

dtD(Stf) + 2γnD(Stf) ≥ nγ/2

n− 1

∫ |∇Stf |2Stf

dσn.

Let

I(f) =

∫

Sn−1

|∇f |2f

dσn

stand for the Fisher information of a probability distribution f on Sn−1. The pre-ceding differential inequality implies

D(f) ≥ nγ/2

n− 1

∫ +∞

0

e−2γntI(Stf) dt.

It is known from the theory of logarithmic Sobolev and hypercontractive inequalities(see for instance [32]) that

(61) I(Sτf) ≤ e−2ρτ I(f), ρ = n− 1

(more generally, on a sphere of radius r, ρ = (n− 1)/r2). Using this and reasoningas in the proof of Theorem 2.1, we end up with

(62) D(f) ≥ nγ/2

γn+ n− 1

∫ +∞

0

I(St′f) dt′ =nγ/2

(γ + 1)n− 1H(f).

This ends the proof of Theorem 6.1. �

We shall conclude this paper on still another open problem. What happens ifone tries to replace Kac’s caricature of a Maxwell gas by a more realistic model, inwhich particles have velocities in R

N (N ≥ 2) and undergo elastic collisions, with acollision kernel of the form B = 1 + |vi − vj|γ (as a first step towards an even morerealistic model with B = |vi − vj|γ) ? More explicitly, consider the case in which

Lnf =n( n

2

)∑

ij

−∫

SN−1

B(vi − vj, σ)[f(v′ij)− f(v)] dσ,

40 CEDRIC VILLANI

where v′ij stands for v, with velocities vi and vj replaced by

v′i =vi + vj

2+|vi − vj|

2σ, v′j =

vi + vj

2− |vi − vj|

2σ,

and B(vi − vj, σ) ≥ 1 + |vi − vj|γ. Then the phase space would be a sphere ofdimension nN − (N + 1), due to the existence of the N + 1 conservation laws,

N∑

i=1

|vi|2 = Nn,N∑

i=1

vi = 0.

We do believe that the same bounds in O(n1−γ/2) still hold true for this model. Thiswould be consistent with both Theorem 2.1 and Theorem 6.1. However, the proofof this fact turns out to require a more delicate geometrical analysis, if one wishesto establish an analogue of Lemma 2.8 in arbitrarily large dimension. This will bethe object of future study.

Added note (January 2005): Remark 2 following Theorem 6.1 is essentially true,but the discussion is much, much more subtle than the statement of this remarksuggests. To begin with, think that it is not a priori clear that the restriction ofthe tensor product f⊗n

0 to the sphere makes any sense if f0 is not continuous. Itturns out that this restriction does make sense if f0 lies in Lp for some p > 1 andhas a bounded moment of order 4 (there is a good reason for this 4, although it

is not trivial at first sight); and then the statement H(f0|M) = limn→∞H(f(n)0 )/n

also holds true. This can be generalized to wider classes of initial data f(n)0 . All this

(and more) is discussed in great detail in a paper that I subsequently wrote alongwith Eric Carlen, Maria Carvalho, Jonathan Le Roux and Michael Loss, Entropyand Chaos in the Kac model.

Acknowledgements: I started to work hard again on Boltzmann’s entropy productionon the occasion of a course that I taught at Institut Henri Poincare (Paris) during thefall of 2001, upon the suggestion of Stefano Olla. The proof of Theorem 2.1 came asan illumination while I was trying to answer some nasty questions by Thierry Bodineau.Thus, both of them have played a crucial role in the genesis of the paper and deserve manythanks. Eric Carlen had an obvious influence on the present work, and was kind enoughto read it carefully and make some constructive remarks. The paper also relies cruciallyon some previous joint works with Laurent Desvillettes and Giuseppe Toscani. Additionalthanks are due to Michael Loss for several enlightening discussions about the subject,and to Laurent Saloff-Coste for pointing out reference [33]. Finally, the support of theEuropean network “Hyperbolic and Kinetic Equations”, contract HPRN-CT-2002-00282,is acknowledged.


Dedication: It is a pleasure to dedicate this paper to Carlo Cercignani, whose influence

on the theory of the Boltzmann equation over the last decades cannot be overestimated.

The results presented here are one instance of the numerous achievements which have been

directly or indirectly triggered by his remarks and ideas.

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C. Villani

UMPA, ENS Lyon

46 allee d’Italie

69364 Lyon Cedex 07

FRANCE

e-mail: [email protected]

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