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ENTROPY PRODUCTIONAND CONVERGENCE TO EQUILIBRIUMC. VILLANI, ENS [e-mail] set of notes was used to complement my short course on the convergenceto equilibrium for the Boltzmann equation, given at Institut Henri Poincar´e inNovember-December 2001, as part of the Hydrodynamic limits program organizedby Stefano Olla and Franc¸ois Golse. The informal style is in accordance withthe fact that this is neither a reference book, nor a research paper. The readercan use my review paper, A review of mathematical topics in collisional kinetictheory, as a reference source to dissipate any ambiguity with respect to notationfor instance. Apart from minor corrections here and there, the main changes withrespect to the original version of the notes were the addition of a final section topresentsomemorerecent developments andopendirections, andthechange of thesign convention for the entropy, to agree with physical tradition. Irene Mazzella iswarmly thanked for kindly typesetting a preliminary version of this manuscript.CONTENTS1. The entropy production problem for the Boltzmann equation 22. Tentative panorama 153. Reminders from information theory 184. Quantitative H theorem 255. Spatially homogeneous Boltzmann equation: state of the art 436. Related topics 497. Wiping out of spatial inhomogeneities 668. Towards exponential convergence? 7911. The entropy production problem for the Boltzmann equation.I shall start with Boltzmann’s brilliant ...

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• Publié par: Voeg
• Nombre de lectures: 13
• Langue: English

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ENTROPY PRODUCTION AND CONVERGENCE TO EQUILIBRIUM

C. VILLANI, ENS LYON [e-mail]

This set of notes was used to complement my short course on the convergence toequilibriumfortheBoltzmannequation,givenatInstitutHenriPoincar´ein November-December 2001, as part of theHydrodynamic limitsprogram organized byStefanoOllaandFranc¸oisGolse.Theinformalstyleisinaccordancewith the fact that this is neither a reference book, nor a research paper. The reader can use my review paper,A review of mathematical topics in collisional kinetic theory, as a reference source to dissipate any ambiguity with respect to notation for instance. Apart from minor corrections here and there, the main changes with respect to the original version of the notes were the addition of a final section to present some more recent developments and open directions, and the change of the sign convention for the entropy, to agree with physical tradition. Irene Mazzella is warmly thanked for kindly typesetting a preliminary version of this manuscript.

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4.

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CONTENTS

The entropy production problem for the Boltzmann equation

Tentative panorama

Reminders from information theory

QuantitativeHtheorem

Spatially homogeneous Boltzmann equation:

Related topics

Wiping out of spatial inhomogeneities

Towards exponential convergence?

1

state of the art

2

15

18

25

43

49

66

79

Notation:

+ boundary conditions.

v ′ = v+ 2 v ∗ + |v− 2 v ∗ ) σ v ′ ∗ = v 2 +v ∗ − |v− 2 v ∗ | σ(σ∈S N−1 ) Think of (v ′  v ′ ∗ ) as possible pre-collisional velocities in a process of elas- tic collision between two particles, leading to post-collisional velocities (v v ∗ )∈IR N ×IR N .

f=f(t x v) f ′ =f(t x v ′ ) f ∗ =f(t x v ∗ ) f ′ ∗ =f(t x v ∗′ )

Q(f f) Z Z

= :=

S N−1

(f ′ f ∗ ′ −f f ∗ )B(v v ∗  σ)dv ∗ dσ − IR N

•

•

2

Physical quantity:B=B(v−v ∗  σ)≥0, the collision kernel (= cross- section times relative velocity) keeps track of the microscopic interaction. It is assumed to depend only on|v−v ∗ |and cosθ, where  | v v − −v v ∗ ∗ | σ   cosθ= (Brackets stand for scalar product.) By abuse of notation I may sometimes writeB(v−v ∗  σ) =B(|v−v ∗ |cosθ).

1. The entropy production problem for the Boltzmann equation. I shall start with Boltzmann’s brilliant discovery that theHfunctional (or negative of the entropy) associated with a dilute gas is nonincreasing with time. To explain the meaning of this statement, let me first recall the model used by Boltzmann. 1.1. The Boltzmann equation: notation and preliminaries

Unknown:f(t x v) =f t (x v)≥0 is a time-dependent probability distri- bution on the phase space Ω x ×IR v N , where Ω x ⊂IR N (N= 2 or 3) is the spatial domain where particles evolve and IR v N is the space of velocities (to be thought of as a tangent space).

Evolution equation:  ∂f (BE)     ∂t +v∇ x f

v *

The picture of collisions is as follows (in IR N v ):

v’ *

θ∋(

Boundary conditions: I shall consider thr

1)

2)

3)

θ

σ

k

v’

v

ee simple cases:

periodic condition: Ω =T N (not really a subset of IR N !), no boundaries.

bounce-back condition: Ω smooth bounded,

f t (x v) =f t (x−v)

specular reflection: Ω smooth bounded,

for

for

x∈∂Ω

f t (x R x v) =f t (x v)x∈∂Ω where  R x v=v−2hn(x) vin(x) n(x) = normal to∂Ω atx

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Local hydrodynamic fields: The following definitions constitute the bridge between the kinetic theory of Maxwell and Boltzmann on one hand, and the classical hydrodynamics on the other hand. Wheneverf(x v) is a kinetic distribution, define the •local densityρ(x) = Z I f(x v)dv R vN Z f(x v)v dv

• local velocity (mean)u(x) = ρ(x) eratureT Z f( N xv)| ρ v ( − x) u(x)| 2 dv •local temp (x) = A simple symmet hat Z IR N ry argument shows tQ(f f)ϕ dv= 0 forϕ=ϕ(v) in Vect (1 v i |v| 2 ) 1≤i≤N , as soon asf=f(v) is integrable enough at large velocities. Thoseϕ’s are calledcollision invariants.

Global conservation laws: Let (f t ) t≥0 be a well-behaved solution of the BE. Then    t d d Z f t (x v)dv dx (conservation= 0 of mass)   dt d Z f t (x v) |v snre(0oc 2 nofoavitticekiney)nerg | 2 d v dx= - Also td d Z f t (x v)v dv dx= 0 in the case of periodic boundary conditions (conservation of momentum). ~ - When Ω has an axis of symmetrykand specular reflection is enforced, then there is an additional conservation law: d d t Z f t (x v)v 0 (k∧n)dv dx of angular momentum) (conservation= 0

(|k|= 1, andn=n(x) is still the normal).

4

me

Normalizations: Without loss of generality I shall assu • Z f t (x v)dvdx= 1 Z f t (x v) |v| 2 dv dx= N 2 2 •and Z f t (x v)v dv dx= 0 in the periodic case

• |Ω|= 1 (|Ω|=N-dimensional Lebesgue measure of Ω)

Moreover, in this course I will not consider the case when Ω has an axis of symmetry and specular boundary condition is imposed. A discussion would have to take into account angular momentum, and consider separately the particular case when Ω is a ball.

1.2.Hfunctional andHTheorem

Let us now introduce Boltzmann’sHfunctional: whenfis a probability distribution on Ω×IR N , define

H(f) = Z

flog dxf dv

This quantity is well-defined in IR∪ {+∞}provided that Z f(x v)|v| 2 dv dx is finite, and will be identified with the negative of the entropy associated withf. The following theorem, essentially due to Boltzmann, will be our starting point.

Theorem 1:Let(f t ) t≥0 be a well-behaved (smooth) solution of the BE (in particular with finite entropy), with one of the boundary conditions discussed above. Then i) d d H t (f t )≤0 one can define a functional. Moreover,DonL 1 (IR v N ), called “entropy production functional”, or “dissipation ofHfunctional”, such that d d t H(f t ) =− Z Ω x D(f t (x ))dx

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ii)

iii)

Assume that the collision kernelB(v−v ∗  σ)is>0for almost all (v v ∗  σ)∈IR 2N ×S N−1 . Letf(x v)be a probability distribution dis- tribution onΩ×IR N , with Z f(x v)|v| 2 dv dx <+∞. Then

Z

D(f(x ))dx= 0 Ω

⇐⇒

fis in local equilibriumiethere exist

functionsρ(x)≥0 u(x)∈IR N  T(x)>0 |v−u(x)| 2 − e 2T(x) such thatf(x v) =ρ(x 2[ ) πT(x)] N2 

Assume that the boundary condition is either periodic, or bounce-back, or specular, and in the latter case assume that the dimension is either 2or3and thatΩhas no axis of symmetry (is not a disk or a cylinder or an annulus or a ball or a shell). Without loss of generality, assume thatfsatisfies the normalizations discussed above. Then (f t ) t≥0 is stationary∀⇐⇒t≥0 Z D(f t (x ))dx= 0 Ω v 2 − ⇐⇒f t (x v  ( ) = e 2 ) 2 N2 ≡M(v)∀t≥0 π

The proof of this theorem is well-known (actually there are several proofs for point (ii), even though not so many), but is is useful to sketch it in order to help understanding refinements to come.

Proof of theorem 1 (sketch):

i)

d d t Z

f t logf t

=

=

=

Z Q(f t  f t )(logf t + 1)− Z (v∇ x f t )(logf t + 1) Z Q(f t  f t ) logf t − Z ∇ x (vf t logf x ) Z Ω×IR N Q(f t  f t ) logf t − Z ∂Ω×IR N [vn(x)]f t logf t

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Under any one of the boundary conditions that we use, the second integral is 0. As for the first one, it can be rewritten as Z Ω Z IR Z (f ′ f ∗ ′ −f f ∗ ) logf B(v−v ∗  σ)dv dv ∗ dσ dx 2N S N−1

By a simple symmetry trick, this is also − 4 1 Z Z 2N Z N−1 (f ′ f ′ ∗ − f f ′ f f ∗′ B dσ dv ∗ dv dx f ∗ ) log f

ΩIRS ∗ which takes the form− Z Ω D(f)dxif one defines theentropy pro- duction functional: D(f)= 1 4 Z IR 2 (f ′ f ′ −f f ∗ f ′ f ′ N ×S N−1 ∗ ) log f f ∗ ∗ B(v−v ∗  σ)dσ dv ∗ dv

ClearlyD(f)≥0 becauseB≥0 and (X−Y) log Y X ≥0 as a conse- quence of log being increasing.

ii) SinceB >the equality means that for (almost)0 almost everywhere, allx∈Ω theL 1 functionf=f(x) satisfies the functional equation of Maxwell–Boltzmann:

(MB) (and also Z ofx’s).

f(v ′ )f(v ′ ∗ ) =f(v)f(v ∗ )

for almost allv v ∗  σ

f(v)(1 +|v| 2 )dv <+∞, up to deletion of a negligible set

Integrate equation (MB) with respect toσ∈S N−1 , to find that 1 f(v)f(v ∗ ) |S N−1 | Z S N−1 f(v ′ )f(v ∗ ′ )dσ = 1 Z S(vv ∗ ) f(α)f(α˜)dα = |S(v v ∗ )|

where

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•S(v v ∗ ) is the collision sphere, centered at v+ 2 v ∗ , with radius |v− 2 v ∗ | •αeis the symmetric ofαwith respect to v 2 +v ∗ . The important point about this average overS(v v ∗ ) is that it only depends uponS(v v ∗ ), whence only upon v 2 +v ∗ and |v− 2 v ∗ | , or (which is equivalent) upon the physically meaningful variables  ∗ total momentum)  m=v+v(

  e =

|v| 2 +|v ∗ | 2 2

Thusf(v)f(v ∗ ) =G(m e).

(total kinetic energy)

Note: In this argument, due to Boltzmann, the Maxwell distribution arises from this conflict of symmetries between the tensor product structure off f ∗ and the dependence ofgupon a reduced set of variables:mande.

Let us continue with the proof of (ii). We first assumefto be smooth (C 1  logarithms, we find Taking, positive).

logf(v) + logf(v ∗ ) = logG(m e) ∂ ∂ v =⇒ ∇logf(v =) + 0∇ v h logG(m e) i =∇ m h logG(m e) i + e∂ ∂ h logG(m e) i v Similarly,∇logf(v ∗ ) =∇ m h logG(m e) i + ∂ ∂e h logG(m e) i v ∗ . So (∇logf)(v)−(∇logf)(v ∗ )/v−v ∗ ∀v v ∗ ∈IR N ×IR N  As a purely algebraic consequence of this (hereN≥2) is crucial), the existsλ∈IR and∈IR N such that

∀v∈IR N ∇logf(v) =λv+

This in turn implies thatfis a Maxwellian distribution.

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re

What happens iffis not smooth? remarked by Desvillettes, the Maxwell– As Boltzmann equation (MB), written with a suitable parametrization of the collisions, is invariant under convolution with a Maxwellian. In fact the following property will be enough for us: if

|v| 2 |v∗ | 2 − − M =( )M δ (v)M δ (v ∗ ) = e 2δ e 2δ δ v v ∗ (2πδ) N2 (2πδ) N2

then fromf(v)f(v ∗ ) =G(m e) we maydeduce

m M δ v ∗ v ∗ f f ∗ ≡M δ v ∗ v ∗ G( e) which is (f∗M δ )(f∗M δ ) ∗ =M δ ∗G(m e)But the convolution byM δ preserves the class of functions of the formG(v+v ∗ |v−v ∗ |). Hence, for allδ >0(f∗M δ )(f∗M δ ) ∗ =G δ (m e)sof∗M δ is a Maxwellian distribution, and we conclude by lettingδ→0 thatfalso is.

Remark: Lions (1994) has a beautiful direct proof that (MB)⇒[fisC ∞ ]. Now let us go on with the proof of Theorem 1 (iii). It is clear from (ii) thatQ(M M Also) = 0.v ∇ x M= 0, soMis a stationary solution. Conversely, let (f t ) t≥0  Frombe a solution which does not produce entropy. (ii) we know that

|v−u(tx)| 2 − 2T(t f t (x v) =ρ(t x 2[ ) Tπ e (t x x ) ) ] N2  So we can plug this into the BE, which reduces to ∂f ∂t +v ∇ x f= 0 since Q(f t  f t ) = 0. Assumingfsmooth>0, we write f 1 ( ∂ ∂ f t +v ∇ x f) = 0 and see that it is in fact a system of polynomial equations inv: 1∂ρ + |v−u| 2 ∂t ∂T +h T v−  ∂u t u ∂ i − 2T N t∂ T∂ ρ ∂t2T 2 v ∇ x ρ|v 2 − T 2 u| 2 v ∇ x T− 2 N v T  ∇ x ρ+ ∇ T x u : [v⊗(−u)] = 0 + +v ρ Let us use the shorthandX ′ =∂X/∂t, and identify powers ofv:

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(0)

(1)

ρ ′ + |u| 2 T ′ − 2T T N ρ2T 2

′ hu u ′ i −= 0 T

T ′ |u| 2 T 2 u+ T u ′ + ∇ ρ ρ + | 2 u T | 2 2 ∇T− 2T N ∇T− T 1 ∇  2  = 0 −

(2) 2 T T ′ 2 δ ij − u j ∂ i T+ 2T 2 u i ∂ j T + ∂ i u j 2 + T ∂ j u i = 0 T (3) 2 ∇ T 2 = 0 For a detailed version of the reasoning which follows, see Desvillettes (1990).

•

•

•

From (3) we have∇ x T= 0, soT=T(t).

Plugging this into (2) we find

′ T ′ I ∂ i u j 2 +∂ j u i =− 2 δ T T ij (in matrix notation∇ sym u=− 2T d) and this expression is independent ofx. From this one easily sees that ∂ ij u i = 0, then∂ ii u j = 0, thenuis an affine map which can be written asu(x) =λx+ Λ∧x+u 0 (λ∈IR, Λ∈IR 3 ) From the boundary conditions, Z Ω (∇ u)dx= Z ∂Ω un= 0 (periodic ⇒∂Ω =∅bounce-back⇒u= 0 on∂Ω, specular⇒un So= 0). λ= 0 andu(x) = Λ∧x+u 0 . This is possible for Λ6= 0 only if Ω has an axis of symmetryk/λand we are considering specular reflection, a case which we have excluded. Sou(x) =u 0 (t), butu 0 can be6= 0 only in the periodic case. To sum up, at this point we know thatu≡0; or u≡u(t) in the periodic case.

Going back to (1): if we knowu≡0, then

∇ρ 0  = ρ

so

ρ=ρ(t);

and if we knowu=u(t), then ∇ ρ ρ is independent ofx, soρ= exp  A(t) x+B(t)  =ρ 0 (t)e A 0 (t)x .

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•

Then, from (0)ρ ′ only depends ont, and so doesρ.

Conclusion:all the cases, we know thatIn •ρ u Tdo not depend onx(case of the torus) •orρ Tdo not depend onxandu≡0 (other cases). But the global conservation laws imply Z ρ= 1, Z ρ |u 2 | 2 + N 2 Z ρT= N 2 (conservation of mass and energy). Also in the periodic case Z ρu= 0 (conservation of momentum). This impliesρ≡1,u≡0,T≡ proof1. The is complete.✷

Theorem 1 is very important! Point (ii) is at the basis of the problem of hydrodynamic limit, while point (iii) is crucial in the problem of trend to equilibrium. We should note that in the particular situation whenf t (x v) = f t (v) [spatial homogeneity,x∈T N ] then (iii) is superfluous.

1.3. What this course is about: convergence to equilibrium Problem of trend to equilibrium: Let be given a solution (f t ) t≥0 of the BE, starting from some initial datumf 0 which is out of equilibrium (i.e., with our normalizations,f 0 6=M). Is it true that

f t −→ t→∞

M

?

Remark 1: A major difference between the problem of hydrodynamic limit and the problem of trend to equilibrium is that in the latter case, one is in- terested in proving that the solution approaches a global equilibrium, while in the former one only expects to come close to a local equilibrium. Accord- ingly, as a general rule the problem of trend to equilibrium is much more sensitive to boundary conditions, than the hydrodynamic limit problem.

Remark 2: Another difference is that one expects shocks occurring asymp- totically as the Knudsen number goes toO, at least in the compressible hy- drodynamic limit, while for fixed Knudsen number there is no known reason for appearance of shocks — at least in a strictly convex domain.

Remark 3: TheHthe only way to attack the problem ofTheorem is not trend to equilibrium. But it seems to be by far the most robust.

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