ENTROPY PRODUCTION AND CONVERGENCE TO EQUILIBRIUM
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ENTROPY PRODUCTION AND CONVERGENCE TO EQUILIBRIUM C. VILLANI, ENS LYON This set of notes was used to complement my short course on the convergence to equilibrium for the Boltzmann equation, given at Institut Henri Poincare in November-December 2001, as part of the Hydrodynamic limits program organized by Stefano Olla and Franc¸ois Golse. The informal style is in accordance with 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. CONTENTS 1. The entropy production problem for the Boltzmann equation 2 2. Tentative panorama 15 3. Reminders from information theory 18 4. Quantitative H theorem 25 5. Spatially homogeneous Boltzmann equation: state of the art 43 6. Related topics 49 7. Wiping out of spatial inhomogeneities 66 8. Towards exponential convergence? 79 1
Voir
- tic collision
- conditions discussed
- cross- section times
- collisional kinetic
- global conservation laws
- local temperature
ENTROPY PRODUCTION AND CONVERGENCE TO EQUILIBRIUM
C. VILLANI, ENS LYON cvillani@umpa.ens-lyon.fr
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 theorya reference source to dissipate any ambiguity with respect to notation, as 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.
1. The entropy production problem for the Boltzmann equation
2. Tentative panorama
3. Reminders from information theory
4. QuantitativeHtheorem
5. Spatially homogeneous Boltzmann equation: state of the art
6. Related topics
7. Wiping out of spatial inhomogeneities
8. Towards exponential convergence?
1
2
15
18
25
43
49
66
79
1. The entropy production problem for the Boltzmann equation. I shall start with Boltzmann’s brilliant discovery that theHlnaoitcnuf (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
:f(t x v) =ft(x v)≥0 is a time-dependent probability distri-bution on the phase space Ωx×IRvN, where Ωx⊂IRN(N= 2 or 3) is the spatial domain where particles evolve and IRNvis the space of velocities (to be thought of as a tangent space).
Evolution equation: ∂f+v∇xf=Q(f f) (BE)∂t :=ZSN−1ZIRN(f′f′∗−f f∗)B(v−v∗ σ)dv∗dσ + boundary conditions.
: •f=f(t x v) f′=f(t x v′) f∗=f(t x v∗) f′∗=f(t x v′∗) •v′=v+2v∗+|v−2v∗)σ v′∗=v2+v∗−|v−2v∗|σ(σ∈SN−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∗)∈IRN×IRN.
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 cosθ=|vv−−vv∗∗| σ (Brackets stand for scalar product.) By abuse of notation I may sometimes writeB(v−v∗ σ) =B(|v−v∗|cosθ).
2
v*
The picture of collisions is as follows (in IRNv):
v’*
θ∋(
θ
σ
k
v’
v
Boundary conditions: I shall consider three simple cases:
1) periodic condition: Ω =TN(not really a subset of IRN!), no boundaries.
2) bounce-back condition: Ω smooth bounded,
3)
ft(x v) =ft(x−v) forx∈∂Ω
specular reflection: Ω smooth bounded,
ft(x Rxv) =ft(x v) forx∈∂Ω whereRn(xvx=)=vno−rm2han(ltx)o∂vΩin(xt)ax
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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) =ZIRNvf(x v)dv Z
f(x v)v dv •local velocity (mean)u(x) =ρ(x) Z
f(x v)|v−u(x)|2dv = •local temperatureT(x)N ρ(x) hows thatZIRNQ(f A simple symmetry argument sf)ϕ dv= 0 forϕ=ϕ(v) in Vect (1 vi|v|2)1≤i≤N, as soon asf=f(v) is integrable enough at large velocities. Thoseϕ’s are calledcollision invariants.
: Let (ft)t≥0be a well-behaved solution of the BE. Then tddZft(x v)dv dx (conservation of mass)= 0 tddZft(x v)|v2|2dv dx of kinetic energy) (conservation= 0 - AlsotddZft(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 dtZft(x v)v0(k∧n)dv dx of angular momentum)= 0 (conservation
(|k|= 1, andn=n(x) is still the normal).
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: Without loss of generality I shall assume •Zft(x v)dvdx= 1Zft(x v)|v2|2dv dx=N2 •andZft(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 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 distribution on Ω×IRN, define H(f) =Z
flogf dv dx
: whenfis a probability
This quantity is well-defined in IR∪ {+∞}provided thatZf(x v)|v|2dv 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.
:Let(ft)t≥0be a well-behaved (smooth) solution of the BE (in particular with finite entropy), with one of the boundary conditions discussed above. Then i)ddtH(ft)≤0 one can define a functional. Moreover,DonL1(IRvN), called “entropy production functional”, or “dissipation ofHfunctional”, such that −ZΩxD(ft(x ))dx ddtH(ft) =
5
ii) Assume that the collision kernelB(v−v∗ σ)is>0for almost all (v v∗ σ)∈IR2N×SN−1. Letf(x v)be a probability distribution dis-tribution onΩ×IRN, withZf(x v)|v|2dv dx <+∞. Then
Z
D(f(x ))dx= 0⇐⇒fis in local equilibriumiethere exist Ω functionsρ(x)≥0 u(x)∈IRN T(x)>0 2 − 2T(x) such thatf(x v) =ρ(x)[2eπT(x)]N2
iii) 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 (ft)t≥0is stationary⇐⇒∀t≥0ZD(ft(x ))dx= 0 Ω 2 − ⇐⇒ft(x v () =e2π)2N2≡M(v)∀t≥0
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.
i) dZ dt
(sketch):
ftlogft= =
=
ZQ(ft ft)(logft+ 1)−Z(v∇xft)(logft+ 1) ZQ(ft ft) logft−Z∇x(vftlogfx) ZΩ×IRNQ(ft ft) logft−Z∂Ω×IRN[vn(x)]ftlogft
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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 ZΩZIR2NZ(f′f′∗−f f∗) logf B(v−v∗ σ)dv dv∗dσ dx SN−1
By a simple symmetry trick, this is also −41ZΩZIR2NZSN−1(f′f′∗−f f∗) logff′ff′∗B dσ dv∗dv dx which takes the form−ZΩD(f)dxif one defines theentropy pro-duction functional: D(f)=14ZIR2N×SN−1(f′f′∗−f f∗) logff′ff′B(v−v∗ σ)dσ dv∗dv ∗ ClearlyD(f)≥0 becauseB≥0 and (X−Y) logXY≥0 as a conse-quence of log being increasing.
ii) SinceB >0 almost everywhere, the equality means that for (almost) allx∈Ω theL1functionf=f(x) satisfies the functional equation of Maxwell–Boltzmann: (MB)f(v′)f(v∗′) =f(v)f(v∗) for almost allv v∗ σ (and alsoZf(v)(1 +|v|2)dv <+∞, up to deletion of a negligible set
ofx’s).
Integrate equation (MB) with respect toσ∈SN−1, to find that f(v)f(v∗) =|SN1−1|ZSN−1f(v′)f(v′∗)dσ =|S(v1 v∗)|ZS(vv∗)α)f( ˜α)dα f(
where
7
atv+v∗with radius|v−v∗| •S(v v∗) is the collision sphere, centered 2 , 2 eis the symmetric ofαwith respect tov+v∗. •α 2 The important point about this average overS(v v∗) is that it only depends uponS(v v∗), whence only uponv+2v∗and|v−2v∗|, or (which is equivalent) upon the physically meaningful variables m∗to e==v|v|+v2+|v∗(|2(otatklergy)ineticennemomlat)mut 2
Thusf(v)f(v∗) =G(m e).
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 (C1, positive). logarithms, we find Taking
logf(v) + logf(v∗) = logG(m e) ∂∂v=⇒ ∇logf(v) + 0 =∇vhlogG(m e)i =∇mhlogG(m e)i+∂∂ehlogG(m e)iv Similarly,∇logf(v∗) =∇mhlogG(m e)i+∂e∂hlogG(m e)iv∗. So (∇logf)(v)−(∇logf)(v∗)/v−v∗∀v v∗∈IRN×IRN As a purely algebraic consequence of this (hereN≥2) is crucial), there existsλ∈IR and∈IRNsuch that ∀v∈IRN∇logf(v) =λv+
This in turn implies thatfis a Maxwellian distribution.
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What happens iff Asis not smooth? remarked by Desvillettes, the Maxwell– 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(v v∗) =Mδe2δe2δ δ(v)Mδ(v∗ (2) =πδ)N2(2πδ)N2
then fromf(v)f(v∗) =G(m e) we maydeduce Mδ v∗v∗f f∗≡Mδv∗v∗G(m e) which is (f∗Mδ)(f∗Mδ)∗=Mδ∗G(m e)But the convolution byMδ preserves the class of functions of the formG(v+v∗|v−v∗|). Hence, for allδ >0(f∗Mδ)(f∗Mδ)∗=Gδ(m e)sof∗Mδis a Maxwellian distribution, and we conclude by lettingδ→0 thatfalso is.
: 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 ∇xM= 0, soMis a stationary solution. Conversely, let (ft)t≥0 Frombe a solution which does not produce entropy. (ii) we know that
|v−u(tx)|2 − ft(x v) =ρ(t x)2[πTe(2tT(txx))]N2 So we can plug this into the BE, which reduces tof∂t∂+v ∇xf= 0 since Q(ft ft Assuming) = 0.fsmooth>0, we write f(1∂t∂f+v ∇xf) = 0 and see that it is in fact a system of polynomial equations inv: 1t∂∂ρ+|v2−T2u|2t∂T∂+hv−∂utu∂i −2T∂TN ρ T ∂t +vρ∇xρ+|v−u|2v ∇xT−2NTv ∇xρ+∇Txu: [v⊗(v−u)] = 0 2T2 Let us use the shorthandX′=∂X/∂t, and identify powers ofv:
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