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Therefore, also the Lagrangian tori obtained with

this procedure have the same dimension of the tori

obtained in the planar case (i.e., four).

The general case Consider the general case follow-

ing the strategy of M Herman as presented by Fe´joz

(2004), to which the reader is referred for complete

proofs and further references.

The symplectic variables used in Fe´joz (2004),to

cope with the spatial planetary (1 þ n)-body prob-

lem (Sun and n planets), are closely related to the

variables defined in eqn [23]. For 1  i  n, let

((L

, G

, 

), (‘

, g

, 

)) denote the Delaunay variables

associated with the two-body system, Sun–ith

planet. Then (as shown by Poincare´) the variables

((

, 

), (

, 

), (p

, q

)), where 

= L

, 

= ‘

þg

þ

and



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2ðL

 G

cosðg

þ 



¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2ðL

 G

sinðg

þ 

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2ðG

 

cos 

¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2ðG

 

sin 

½26

are symplectic and analytic near circular, non-

coplanar motions (see, e.g., Biasco et al. (2003)).Let

nbp

:¼H

ð0Þ

ðÞþ"H

ð1Þ

ð;;;;p; qÞ½27

denote the Hamiltonian (eqn [8]) expressed in terms

of the Poincare´ symplectic variables ((, ), (, ),

(p, q)), =(

, ..., 

), etc.

As the number of the planets increases, the

degeneracies become stronger and stronger. Further-

more, a clean reduction, such as the reduction of the

nodes, is no more available if n > 2. To overcome

these problems Herman proposed a new approach,

which is described below.

Instead of Kolmogorov’s nondegeneracy assump-

tion – which says that the frequency map [13]

I !!(I) is a local diffeomorphism – one may

consider weaker nondegeneracy conditions. In

particular, in Fe´joz (2004), one considers non-

planar frequency maps. A smooth curve u 2 A !

!(u) 2 R

, where A is an open nonempty interval,

is called ‘‘nonplanar’’ at u

2 A if all the u-derivatives

up to order (d  1) at u

, !(u

), !

), ..., !

(d1)

)

are linearly independent in R

; a smooth

map u 2 A  R

!!(u) 2 R

, p  d, is called

nonplanar at u

2 A if there exists a smooth

curve ’ :

A !A such that !  ’ is nonplanar at t

A with ’(t

) = u

. A S Pyartli has proved (see, e.g.,

Fe´joz (2004)) that if the map u 2 A  R

!!(u) 2 R

is nonplanar at u

, then there exists a neighborhood

B  A of u

and a subset C  B of full Lebesgue

measure (i.e., meas(C) = meas(B)) such that !(u)is

Diophantine for any u 2 C. The nonplanarity condi-

tion is weaker than Kolmogorov’s nondegeneracy

conditions; for example, the map

!ðIÞ:¼ @

þ I



¼ I

þ 2I

þ I

; I

; 1



violates both Kolmogorov’s nondegeneracy and the

isoenergetic nondegeneracy conditions but is non-

planar at any point of the form (I

, 0, 0, 0), since

!(I

,0,0,0)= (I

, I

, 1) is a nonplanar curve (at

any point).

As in the three-body case, the frequency map is

that associated with the averaged secular

Hamiltonian

sec

:¼H

ð0Þ

ðÞþ"



ð1Þ



ð1Þ

ð;;;p; qÞ:¼

ð1Þ

d

ð2Þ

½28

which has an elliptic equilibrium at  =  = p = q = 0

(as above,  is regarded as a parameter). It is a

remarkably well-known fact that the quadratic part



(1)

does not contain ‘‘mixed terms,’’ namely,



ð1Þ



ð1Þ

þ " Q

pln

   þQ

pln

   þQ

spt

p  p



þQ

spt

q  q þ O



½29

where the function



(1)

and the symmetric matrices

pln

and Q

spt

depend upon  while O

denotes

terms of order 4 in (, , p, q). The eigenvalues of the

matrices Q

pln

and Q

spt

are the first Birkhoff

invariants of



(1)

(with respect to the symplectic

variables ( , , p, q)). Let 

, ..., 

and &

, ..., &

denote, respectively, the eigenvalues of Q

pln

and

spt

; then the frequency map for the (1 þ n)-body

problem will be defined as (recall eqn [18] )

 !ð

!; "Þ½30

with

! :¼





; ...;







:¼ð; &Þ :¼ð

; ...;

Þ; ð&

; ...;&

ÞðÞ

½31

Herman pointed out, however, that the frequencies

 and & satisfy two independent linear relations,

namely (up to renumbering the indices),

¼ 0;

i¼1

ð

þ &

Þ¼0 ½32

which clearly prevents the frequency map to be

nonplanar; the second relation in eqn [32] is usually

KAM Theory and Celestial Mechanics 197

called ‘‘Herman resonance’’ (wh ile the first relation

is a well-known consequence of rotation invariance).

The degeneracy due to rotation invariance may

be easily taken care of by considering (as in the

three-body case) the (6n  2)-dimensional invariant

symplectic manifold M

ver

, defined by taking the

total angular momentum C to be vertical, that is,

C  k

= 0 = C  k

. But, when n > 2, Jacobi’s reduc-

tion of the nodes is no more available and to get rid

of the second degener acy (Herm an’s resonance), the

authors bring in a nice trick, originally due – once

more! – to Poincare´. In place of considering H

nbp

restricted on M

ver

,Fe´joz considers the modified

Hamiltonian



nbp

:¼H

nbp

þ C

; C

:¼ C  k

¼jCj½33

where  2 R is an extra artificial parameter. By an

analyticity argument, it is then possible to prove that

the (rescaled) frequency map

ð;Þ!ð

!; 

; ...;

; ...;&

n1

Þ2R

3n1

is nonplanar on an open dense set of full measure

and this is enough to find a positive measure set of

Lagrangian maximal (3n  1)-dimensional invariant

tori for H



nbp

; but, since H



nbp

and H

nbp

commute, a

classical Lagrangian intersection argument allo ws

one to conclude that such tori are invariant also for

nbp

yielding the complete proof of Arnol’d’s

theorem in the general case. Notice that this

argument yields (3n  1)-dimensional tori, which in

the three-body case means five dimensional. Instead,

the tori found in the sect ion ‘‘The spatial three -body

proble m’’ are four dimensi onal. The point is that

in the reduced phase space, the motion of the

nodeline – denoted as

(t)ineqn [25] – does not

appear.

We conclude this discussion by mentioning that

the KAM theory used in Fe´joz (2004) is a modern

and elegant function-theoretic reformulation of the

classical theory and is based on a C

local inversion

theorem (F Sergeraer t and R Hamilton) on ‘‘tame’’

Frechet spaces (which, in turn, is related to the

Nash–Moser implicit function theorem; see Bost

(1984–85)).

Lower Dimensional Tori

The maximal tori for the many-body problems

described above are found near the elliptic equilibria

given by the decoupled Keplerian motions. It is

natural to ask what happens of such elliptic

equilibria when the interaction among planets is

taken into account. Even though no complete

answer has yet been give n to such a question, it

appears that, in general, the Kepler ian elliptic

equilibria ‘‘bifurcate’’ into elliptic n-dimensional

tori. This section presents a short and nontechnical

account of the existing results on the matter (the

general theory of lower-dimensional tori is, mainly,

due to J K Moser and S M Graff for the hyperbolic

case and V K Melnikov, H Eliasson, and S B Kuksin

for the technically more difficult elliptic case; for

references, see, e.g., Chierchia et al. (2004)).

The normal form of a Hamiltonian admitting an

n-dimensional elliptic invariant torus T of energy E,

proper frequencies

! 2 R

, and ‘‘normal frequen-

cies’’  2 R

in a 2d-dimensional phase space with

d = n þ p is given by

N :¼ E þ

!  y þ

j¼1





þ 

½34

Here the symplectic form i s given by dy ^ dx þ

d ^ d, y 2 R

, x 2 T

,(, ) 2 R

; T is then given

by T := {y = 0}  { =  = 0}. Under suitable assump-

tions, a set of such tori persists under the effect of a

small enough perturbation P(y, x, , ). Clearly, the

union of the persistent tori (if n < d)formsasetof

zero measure in phase space; however, in general,

n-parameter families persist.

In the many-body case considered in this article,

the proper frequencies are the Keplerian frequencies

given by the map  !

!()(eqn [31]), which is a

local diffeomorphism of R

. The normal frequencies

, instead, are proportional to " and are the first

Birkhoff invariants around the elliptic equilibria as

discussed above. Under these circumstances, the main

nondegeneracy hypothesis needed to establish the

persistence of the Keplerian n-dimensional elliptic tori

boils down to the so-called Melnilkov condition:



6¼ 0 6¼ 

 

; 8j 6¼ i ½35

Such condition has been checked for the planar

three-body case in Fe´joz (2002), for the spatial

three-body case in Biasco et al. (2003) and for the

planar n-body case in Biasco et al. (2004). The

general spatial case is still open: in fact, while it is

possible to establish lower-dimensional elliptic tori

for the modified Hamiltoni an H



nbp

in [33], it is not

clear how to conclude the existence of elliptic tori

for the actual Hamiltonian H

nbp

since the argument

used above works only for Lagrangian (maximal)

tori; on the other hand, the direct asymptotics

techniques used in Biasco et al. (2003) do not

extend easily to the general spatial case.

Clearly, the lower-dimensional tori descr ibed in

this section are not the only ones that arise in

n-body dynami cs. For more lower-dimensional tori

in the planar three-body case, see Fe´joz (2002).

198 KAM Theory and Celestial Mechanics

Physical Applications

The above results show that, in principle, there may

exist ‘‘stable planetary systems’’ exhibiting quasiper-

iodic motions arou nd coplanar, circular Keplerian

trajectories – in the Newtonian many-body approx-

imation – provided the masses of the planets are

much smaller than the mass of the central star.

A quite different question is: in the Newtoni an

many-body approximation, is the solar system or,

more in generally, a solar subsystem stable?

Clearly, even a precise mathematical reformula-

tion of such a question might be difficult. However,

it might be desirable to develop a mathematical

theory for important physical model s, taking into

account observed parameter values.

As a very preliminary step in this direction, consider

one of the results of Celletti and Chierchia (see Celletti

and Chierchia (2003), and references therein).

In Celletti and Chierchia (2003), the (isolated)

subsystem formed by the Sun, Jupiter, and asteroid

Victoria (one of the main objects in the Asteroidal

belt) is considered. Such a system is modeled by an

order-10 Fourier truncation of the RPC3BP, whose

Hamiltonian has been described in the section

‘‘Kolmogorov’s theorem and the RPC3BP (1954).’’

The Sun–Jupiter motion is therefore approximated by

a circular one, the asteroid Victoria is considered

massless, and the motions of the three bodies are

assumed to be coplanar; the remaining orbital

parameters (Jupiter/Sun mass ratio, which is approx-

imately 1/1000; eccentricity and semimajor axis of the

osculating Sun–Victoria ellipse; and ‘‘energy’’ of the

system) are taken to be the actually observed values.

For such a system, it is proved that there exists an

invariant region, on the observed fixed energy level,

bounded by two maximal two-dimensional Kolmo-

gorov tori, trapping the observed orbital parameters of

the osculating Sun–Victoria ellipse.

As mentioned above, the proof of this result is

computer assisted: a long series of algebraic compu-

tations and estimates is performe d on computers,

keeping a rigorous track of the numerical errors

introduced by the machines.

Acknowledgments

The author is indebted t o J Fe´ joz for explaining

his work on Herman’s proof of Arnol’d’s theorem

prior to publication. The author is also grateful

for the collaborations with his colleagues and

friends L Biasco, E Valdinoci, and especially A

Celletti.

See also: Averaging Methods; Diagrammatic

Techniques in Perturbation Theory; Gravitational

N-Body Problem (Classical); Hamiltonian Systems:

Stability and Instability Theory; Hamilton–Jacobi

Equations and Dynamical Systems: Variational Aspects;

Korteweg–de Vries Equation and Other Modulation

Equations; Stability Problems in Celestial Mechanics;

Stability Theory and KAM.

Further Reading

Arnol’d VI (1963a) Proof of a Theorem by A. N. Kolmogorov

on the invariance of quasi-periodic motions under small

perturbations of the Hamiltonian. Russian Mathematical

Survey 18: 13–40.

Arnol’d VI (1963b) Small denominators and problems of stability

of motion in classical and celestial mechanics. Uspehi

MatematiI

ceskih Nauk 18(6(114)): 91–192.

Arnol’d VI (ed.) (1988) Dynamical Systems III, Encyclopedia of

Mathematical Sciences, vol. 3. Berlin: Springer.

Biasco L, Chierchia L, and Valdinoci E (2003) Elliptic two-

dimensional invariant tori for the planetary three-body

problem. Archives for Rational and Mechanical Analysis

170: 91–135.

Biasco L, Chierchia L, and Valdinoci E (2004) N-dimensional

invariant tori for the planar (N þ1)-body problem. SIAM Journal

on Mathematical Analysis, to appear, pp. 27 (http://www.mat.

uniroma3.it/users/ch ierchia/WW W/engli sh_ve rsion.html ).

Bost JB (1984/85) Tores invariants des syste` mes dynamiques

hamiltoniens. Se´minaire Bourbaki expose 639: 113–157.

Celletti A and Chierchia L (2003) KAM stability and celestial

mechanics. Memoirs of the AMS, to appear, pp. 116. (http://

www.mat.uniroma3.it/users/chierchia/WWW/english_version.

html).

Chierchia L and Qian D (2004) Moser’s theorem for lower

dimensional tori. Journal of Differential Equations 206:

55–93.

Fe´joz J (2002) Quasiperiodic motions in the pl anar three-body

problem. Journal of Diffe rential Equati ons 183( 2):

303–341.

Fe´joz J (2004) De´monstration du ‘‘the´ore` me d’Arnol’d’’ sur la

stabilite´ du syste` me plane´taire (d’apre` s Michael Herman).

Ergodic Theory & Dynamical Systems 24: 1–62.

Herman MR (1998) De´monstration d’un the´ore` me de V.I.

Arnol’d. Se´minaire de Syste`mes Dynamiques and manuscripts.

Kolmogorov AN (1954) On the conservation of conditionally

periodic motions under small perturbation of the Hamilto-

nian. Doklady Akademii Nauk SSR 98: 527–530.

Laskar J and Robutel P (1995) Stability of the planetary three-

body problem. I: Expansion of the planetary Hamiltonian.

Celestial Mechanics & Dynamical Astronomy 62(3):

193–217.

Robutel P (1995) Stability of the planetary three-body problem.

II: KAM theory and existence of quasi-periodic motions.

Celestial Mechanics & Dynamical Astronomy 62(3):

219–261.

Salamon D (2004) The Kolmogorov–Arnol’d–Moser theorem.

Mathematical Physics Electronic Journal 10: 1–37.

Siegel CL and Moser JK (1971) Lectures on Celestial Mechanics.

Berlin: Springer.

KAM Theory and Celestial Mechanics 199

Kinetic Equations

C Bardos, Universite

de Paris 7, Paris, France

Introduction

In most physical cases, the evolution of a system of N

indistinguishable interacting particles X

= (x

, x

, ...,

) with velocities V

= (v

, v

, ..., v

) is described by

a Hamiltonian system

@HðX

; V

¼

@HðX

; V

½1

in the phase space R

 R

.WhenN becomes

large, it is natural to consider replacing the above

discrete phase s pace by a continuous phase space

of dimension 1  d  3, R

 R

and to introduce

ameasuref (x, v, t) that describes the density of

particles which, at the point x 2 R

and at time t,

have velocity v.Thismeasuremayalsobe

interpreted a s a generalization of the empirical

measure



ðtÞ¼

1iN



ðtÞ;v

ðtÞ

defined in the phase space R

 R

by the above

system of N particles. In this way, one constructs a

link between the microscopic and the macroscopic

descriptions. The macroscopic physical quantities

are, for instance, the first moments of this density:

ðx; tÞ¼

f ðx; v; tÞdv ðdensityÞ

ðx; tÞuðx; tÞ¼

vf ðx; v; tÞdv ðmomentumÞ

ðx; tÞEðx; tÞ¼

jvj

f ðx; v; tÞ dv ðenergyÞ

Kinetic theory studies the intermediate stage shown

in Figure 1.

Its first successes were related to classical thermo-

dynamics and in particular to the molecular hypoth-

esis. The contributions of Maxwell (1860, 1872)

and of Boltzmann (1867) led to the ‘‘Boltzmann’’

equation, described in the companion article of

Mario Pulvirenti ( see Boltzmann Equation (Classical

and Quantum)). In 1905, Lorentz used the same

point of view to describe the motion of electrons in a

metal. However, the different physical context leads

to some basic differences between the Boltzmann

equation and the Lorentz equ ation. The Boltzmann

equation is derived under the assumption that the

driving forces result from collisions between pairs of

molecules. Therefore, the problem is nonlinear with

a quadratic nonlinearity. In the Lorentz model the

driving force is the interaction of the electrons with

the atoms of the metal, which remain fixed.

Collisions between electrons are ignored, so that

the Lorentz equation is linear.

The most general form of a kinetic equation is as

follows:

f ðx; v; tÞþr

r

f ðx; v; tÞ

r

r

f ðx; v; tÞ¼Cðf Þ½2

The term C(f ) represents the effect of interactions

either between particles or with the background.

Without this term, the eqn [2] is reduced to the

classical Liouville equation

f ðx; v; tÞþr

r

f ðx; v; tÞ

r

r

f ðx; v; tÞ¼0 ½3

which says that the function f is transported by the

flow of the Hamiltonian H

(x, v). This Hamiltonian

depends on the model and may involve the unknown

function f itself. In the simplest case H(x, v) = jvj

=2,

eqn [3] and its solutions are given by

f ðx; v; tÞþv r

f ðx; v; tÞ¼0

f ðx; v; tÞ¼f ðx  vt; v ; 0Þ½4

Nowadays kinetic equations appear in a variety of

sciences and applications, such as astrophysics,

aerospace engineering, nuclear engineering, particle–

fluid interactions, semiconductor technology, social

sciences, and biology, for example in chemotaxis

and immunology.

They are used first to model phenomena and then

to obtain a qualitative and quantitative description

of situations involving sufficiently many particles so

as to prohibit any computation at the level of

particles, and yet the medium is still too rarefied to

allow the use of macroscopic equations. As detailed

in the next section, a macr oscopic description

requires that the function f (x, v, t) be close to local

thermodynamical equilibrium. For classical and

quantum Boltzmann equations (see Boltzmann

Hamiltonian Systems

Kinetic equations Macroscopic equations

→→

Figure 1 Illustration of the role of kinetic equations in linking

microscopic and macroscopic properties.

200 Kinetic Equations

Equation (Classical and Quantum)) these equilibria

are either Maxwellian, Bose–Einstein, or Fermi–

Dirac distributions.

Several effects, especially the influence of the

boundary, may prevent the system from reaching

local thermodynamical equilibr ium and, therefore,

even in macroscopic descriptions, kinetic equations

may still be used to take into account the eff ect of

the boundary. In this case, the term ‘‘Knudsen

boundary layer’’ is currently used.

Finally, one should keep in mind that there exist

some macroscopic phenomena which cannot be

deduced from the corresponding microscopic phys-

ics by the mediati on of a kinetic equation. Once

again, returning to the companion article (see

Boltzmann Equation (Classical and Quantum)) one

observes that, since the only equilibria are Maxwel-

lian, the macroscopic equations are those describing

perfect gases. A real gas with a nontrivial van der

Waals law is ‘‘too dense’’ to be explained by this

theory. The alternative seems to go directly from the

microscopic direction to the macroscopic descrip-

tion. This is a subject which is still under investiga-

tion and for which the reader may consult Olla et al.

(1993).

Kinetic Equations Entropy

and Irreversibility

At the level of particles, the basic laws of physics are

reversible. Yet these same laws are not reversible

when seen at the level of a macroscopic description.

This lack of reversibility is measured by the decay of

entropy (mathematicians prefer convex funct ions;

therefore, the mathematical entropy considered in

this contribution is the negative of the physical

entropy, and with irreversibility it decays). The

kinetic equations lie in between, as shown in

Figure 1; the decay of entropy should appear along

one of the two arrows of this diagram.

Since the appearance of irreversibility is related to

loss of information and averaging, it should be

driven by a ‘‘mixing’’ process.

In general two mechanisms are responsible for

such effects:

1. an ergodic or a relaxation mechanism by which a

process averages itself; and

2. the introduction of some external random param-

eter. Observable quantities are then defined as

averages over that parameter.

It seems important to compare these two ‘‘pro-

cesses.’’ This will be illustrated below with the most

classical examples of the theory.

The Diffusion Limit for the Neutron Transport

Equation

Equations very similar to the one introduced by

Lorentz are used to describe the interaction of neutrons

with atoms in a nuclear reactor: this is the reason why

these types of equations are often called neutron

transport equations. An important issue is the deriva-

tion of a macroscopic diffusion equation. Assuming

that neutrons are not subject to acceleration effects,

considering the problem with constant modulus of

velocity (jvj= 1), introducing a ‘‘small’’ parameter 

which here corresponds to the absorption of the

medium, one can study the following simplified model:

@



þ v r



ðxÞ









j¼1

kðv; v

Þf



ðv

Þdv



¼ 0 ½5

In [5] one assumes, for the kernel k(v, v

), the

following properties:

8v; v

; kð v ; v

Þ¼kðv

; vÞ; 0 < kðv; v

j¼1

kðv; v

Þdv ¼ 1 ½6

and denotes by K the operator

f 7!Kf ¼

j¼1

kðv; v

Þf ðv

Þdv

In the simplest case (say without boundary) eqn [5]

is well-posed both for positive and negative time

but hy pothesis [6] has the following important

consequences:

1. For positive time, it defines, for each >0, a

contraction semigroup in any L

space and, there-

fore, the sequence of solutions or a subsequence

thereof converges, say weakly, to a limit f (x, v, t).

2. One also observes that v 7!1 is (up to a multi-

plicative constant) the only solution of the equation

f  Kf ¼ f ðvÞ

j¼1

kðv; v

Þf ðv

Þdv ¼ 0 ½7

Therefore, the 

1

in front of the collision term

forces the limit f (x, v, t) to be independent of v.

In this simple problem, this is the thermodyna-

mical equilibrium.

Dividing by  and integrating over jvj= 1 gives the

relation

jvj¼1



ðx; v; tÞdv

þr



jvj¼1



ðx; v; tÞdv ¼ 0 ½8

Kinetic Equations 201

Now using the Fredholm alternative implies the

existence and uniqueness of a function v 7!(v) such

that

ðvÞ

j¼1

kðv; v

Þðv

Þdv

¼ v;

j¼1

ðv

Þdv

¼ 0 ½9

Multiply eqn [5] by (v) and integrate over jvj= 1to

obtain

lim

!0

ðxÞ



jvj¼1

ððI  KÞÞðvÞf



ðx; v; tÞdv

¼ lim

!0

ðxÞ



jvj¼1

ðvÞðI  KÞf



ðx; v; tÞdv

¼lim

!0

jvj¼1

ðvÞvf



ðx; v; tÞdv ½10

Since the operator (I  K) is self-adjoint non-

negative, with 0 as the leading eigenvalue, the

matrix

D ¼

jvj¼1

ðvÞvdv

jvj¼1

ðvÞðI  KÞðvÞdv

is positive definite, and one finally obtains the

diffusion equation

f r

ðxÞ



¼ 0 ½11

The above derivation is an example of what is called

the ‘‘moments method.’’ It is implicit even in the

papers of Maxwell. It has been systematically used

in several domains:



To understand the relation between the Boltzmann

equation and the Euler and Navier–Stokes

equations (Golse 2005);



To compute the critical size of a nuclear assembly.

One shows that this size is well approximated by

the size of the domain for which the Laplacian,

with appropriate boundary conditions, has lead-

ing eigenvalue 0. It is for the spectral analysis of

this problem that the averaging lemma (see the

section ‘‘Some speci fic math ematical tools ’’) was

derived.



To analyze the macroscopic limit for the solution

of the radiative transfer equations, which describe

the propagation of the intensity of photons in a

large class of phenomena ranging from stellar

atmospheres to the cooling of glass, including

optical tomography in biomedical imaging. In a

simplified form, the so-called ‘‘grey model,’’ these

equations can be reduced to

@



ðx; v; tÞþv r



ðx; v; tÞ







4

j¼1



ðx; v

; tÞÞdv





ðx; v; tÞ



4

j¼1



ðx; v

; tÞÞdv



¼ 0 ½12

In contrast to the previous example, the problem

is, in many cases, nonlinear. The opacity  is a

positive function that depends on the intensity I



through

Iðx; tÞ¼

4

j¼1



ðx; v

; tÞdv

and which goes to 1 with



going to zero. The

moments method can be applied with the aver-

aging lemma, and one shows that the limit of I



a function that is independent of v and satisfies

the following degenerate parabol ic equation:

I r

3ðIÞ



¼ 0 ½ 13

This equation is similar to the one obtained in the

description of porous media and contains the

following information: for initial data I(x, 0) with

compact support, in contrast to the behavior of

solutions of the standard diffusion equation, the

solution I(x, t) remains compactly supported in x.

The boundary of this support is the thermal front

and for a finite time, up to saturation (by water in

porous media, by reacted deuterium in laser-

confined fusion), this front remains fixed.

What made the analysis of the above macroscopic

limit simple was the existence of an >0 dependent

process which, for vanishing , forces the solution to

converge to a ‘‘thermodynamical’’ equilibrium. The

irreversibility was already present in the first arrow

of Figure 1. This is what made the analysis of the

second arrow simple. The subtleties of the appear-

ance of the irreversibility in the first arrow may be

well explained by the next examples.

The Linear Billiard Model

In the absence of an external electric field, the model

proposed by Lorentz could be viewed as a limit of a

system of particles evolving freely between spherical

obstacles and reflecting on these obstacles according

to the law of geometric optics. Along these lines,

two types of results have been proved in two space

variables.

202 Kinetic Equations

In 1973, Gallavotti considered the case where the

obstacles are randomly spaced under a Poisson

configuration and proved the following theorem:

Theorem 1 Consider obstacles(balls) of radius 

and center c

. Assume that the probability of finding

exactly N such obstacles in a bounded measurable

set   R

is given by the ‘‘Poisson law’’

Pðdc

Þ¼e





jj





 dc

½14

with

¼ c

; c

; ...; c

and 







½15

Denote by E



the expectation with respect to the

above Poisson distribution. For given  and c

introduce

;

¼ R

1iN

fjx  c

jg½16

and f

, 

, the solution of the problem

;

ðx; v ; tÞþv r

;

ðx; v; tÞ¼0

in O

;

 S

½17

with specular reflection on the boundary and

v-independent initial data:

;

ðx; v; 0Þ¼ðxÞ in O

;

 S

½18

Then



ðx; t; Þ¼E



½f

;

½19

converges weakly for t  0 to the solut ion of the

transport equation

f ðx; v; tÞþv rf ðx; v; tÞþ



2f ðx; vÞ



f ðx; v

Þjv  v

jdv



¼ 0 ½20

f ðx; v; 0Þ¼ðxÞ in R

 S

½21

The situation is completely different when the

obstacles are periodically spaced, a situation which

seems closer to Lorentz’s original idea. Golse (2003)

(and previous contributions quoted in this article)

obtained the following result:

Theorem 2 Assume that the obstacles are periodi-

cally spaced and conveniently scaled , defining the

domain



¼ R

j2Z

fx; jx  jj

g½22

Then there exists a family of continuous uniformly

bounded ini tial data such that no subsequence

extracted from the family of solutions of



þ v r



¼ 0 in O



 S

½23

with specular reflections on the boundary, converges

to solutions of equations of the type [20].

This pathology is related to the existence of

particles that can travel freely for a very long time

before meeting the obstacles, and the proof with

some arithmetic (Diophantine approximations and

continued fractions) relies on the analysis of such

trajectories.

A comparison between the Theorems 1 and 2

shows that the ergodic property of the free flow on

the periodic lattice is not strong enough to lead to a

collisional kinetic equation unless some complemen-

tary randomness is introduced.

The examples of this section should be compar ed

with the rigorous derivation of the Boltzmann

equation by Lanford (see Boltzmann Equation

(Classical and Quantum)). The reader should

observe that this derivation corresponds to the

same type of scaling (finite mean free path).

However, no extra randomness is needed in this

case. The proof uses the fact that configurations

leading only to a finite number of binary collisions

are of full measure. This corresponds to an

ergodicity property which is enforced by the fact

that the problem is genuinely nonlinear.

Mean-Field Scaling and Vlasov Equations

The neutron transport equation is devoted to the

interaction with obstacles and the Boltzmann

equation to binary collisions. A simpler situation

from the mathematical point of view corresponds

to the case where each particle is under the action

of the average of all other particles. Then the name

‘‘mean field limit’’ is used. The simplest example is

the derivation of a Vlasov-type equation from a

system of N classical particles interacting w ith a C

potential V(jxj). The following Hamiltonian is

used:

Hðx

; ...; x

; v

; ...; v

1kN

1l6¼kN

Vðjx

 x

jÞ ½24

and the name mean-field scaling is related to the

factor N

1

before the potential. Assuming that the

particles are undistinguishable, one introduces

the joint probability density F

 F

, ..., x

Kinetic Equations 203

, ..., v

) in the N-particle phase space, which

satisfies the Liouville equation

þfH

; F

g:¼@

1kN



1l6¼k N

ðVðjx

 x

ÞÞ

r



¼ 0 ½25

From [25], with the notations

¼ðx

; ...; x

Þ; V

¼ðv

; ...; v

¼ðx

nþ1

; ...; x

Þ; V

¼ðx

nþ1

; ...; x

one deduces an infinite hierarchy of equations for

the marginals

ðX

; V

; tÞ¼

ðX

; V

; tÞdX

for 1  n  N; F

 0 for N < n :

ðX

; V

; tÞþ

1in

ðX

; V

; tÞ



1i<jn

Vðjx

 x

jÞF

ðX

; V

; tÞ





N  n



1in

Vðjx

 x



jÞ

 F

nþ1

ðX

; V

; x



; v



; tÞdx





¼ 0 ½26

Letting N go to infinity, one obtains ‘‘formally,’’ for

the distribution functions,

¼ lim

N!1

the Vlasov hierarchy:

ðX

; V

; tÞþV

r

ðX

; V

; tÞ



1in



Vðjx

 x



jÞ

 F

nþ1

ðX

; V

; x



; v



; tÞdx





¼ 0 ½27

Observe that for any density F(x, v, t) that satisfies

Fðx; v; t Þdx dv ¼ 1; Fðx; v; tÞ0 ½28

and is a solution of the V potential Vlasov equation:

Fðx; v; t Þþv r

Fðx; v; t Þ



Vðjx  x



jÞFðx



; v



Þdx





r

Fðx; v; t Þ¼0 ½29

the factorization formula

ðX

; V

; tÞ¼

1in

Fðx

; v

; tÞ½30

defines a solution of the above Vlasov hier archy.

A uniqueness argum ent implies that any solution

of the Vlasov hierarchy which is factorized at time

zero will remain factorized at any subsequent time.

Such a property, also observed for the hierarchy

leading to the Boltzmann equation, is called the

propagation of chaos. To make the proof rigorous,

one has to analyze the limiting process in the

hierarchy and prove the uniqueness of the solution

of the infinite hierarchy. For a smooth potential, this

has been done by Braun and Hepp in 1977 and by

Spohn in 1981. An interesting approach consists,

following Dobrushin, in introducing the Wasserstein

distance; see Golse (2003) for a detailed exposition.

In the case of the Vlasov–Poisson equation [29]

with V(jxj) = 1=4jxj the potential turns out to be

too singular for the above derivation. In particular,

the corresponding solution of the N-particle pro-

blem is not uniformly defined. However, for the

corresponding equation (and for variants thereof,

including the effect of the magnetic field, the

Vlasov–Maxwell system) a series of mathematical

results concerning existence and stability of solu-

tions have been obtained. An excellent recent

exposition of these results can be found in the

book of Glassey (1996).

Equation [29] as well as the original system turns

out to be fully reversible. Neither irreversibility nor

averaging has appeared in the limit process which

corresponds to the first arrow of Figure 1; this is due

to the ‘‘weak coupling.’’ Therefore, irreversibility

should now appear on the second arrow. Integrating

eqn [29] with respect to v gives the relation (often

called Fick’s law):

ðx; tÞþr

vFð x; v; t Þdv ¼ 0 ½ 31

But now expressing the curre nt j =

vF(x, v, t)dv in

terms of macroscopic variables turns out to be a

difficult issue in the absence of a ‘‘relaxation’’ effect.

Up to now there has been no derivation of such

macroscopic equations from first principles.

The same type of problems exist for the two-

dimensional Euler equation, which is in some sense

very similar to the Vlasov equation. It has been

observed that these equations develop for ‘‘turbulent

initial data’’ a kind of ‘‘mixing process’’ leading to

coherent structures that would play the role of

thermodynamical equilibrium (in the absence of

relaxation). The Jupiter red spot is the most

204 Kinetic Equations

well-known example of such a structure. These

coherent structures are obtained by maximizing an

entropy which does not come directly from the

dynamics but which is inspired by similar problems

in statistical mechanics. Finally, one has to take into

account in this construction the existence of an

infinite set of conserved quantities: for each regular

function G, vanishing at infinity, one has

GðFðx; v; t ÞÞdx dv ¼ 0

This approach was already started by Onsager in 1945

and pursued by many scientists. A recent reference is

the article of Chavanis and Sommeria (1998).

Derivation of Kinetic Equations from the

Schro¨ dinger Equation

Oscillatory solutions of the Schro¨ dinger equation,

with wavelength of the order of the Planck constant,

tend to behave like particles. This is described in

detail by different tools of high-frequency approxi-

mation. In particular, the limit of the Wigner

transform of the density (x, t) 

(y, t):

Wðx;;tÞ¼

ð2Þ

iy

x þ

hy

; t





x 

hy

; t



dy ½32

is a solution of a Liouville equation. Therefore, one

should expect that in the presence of ‘‘many’’

obstacles (‘‘many potentials’’) the limit should be

given by a kinetic equation. As shown by the

previous section the introduction of randomness

seems compulsory in reachin g this goal.

Consider a big cube =

of size L in R

. Let

! = (x



),  = 1, 2, ..., N denote the configuration of

random obstacles distributed uniformly in . The

density of obstacles is  = N=L

and the expectation

with respect to this uniform measure is denoted by

E :¼

1N



d





With V(jxj) a smooth, short-range potent ial, the

random potential created by the obstacles is

ðxÞ¼

1N

Vðjx  x



jÞ

then one of the typical results (low-density limit,

which corresponds to the quantum version of

Gallavotti classical result) obtained, reads as follows:

Theorem 3 (Erdo¨ s and Yau 1988) Assume that the

density of obstacles is  = 

 with a fixed 

Denote by



(t) the solution of the Schro¨ dinger

equation



¼



þ V



½33

with initial condition localized and oscillating at the

scale , that is, with h and S smooth



ð0Þ¼

3=2

hðxÞexp i

SðxÞ





½34

Consider the density matrix 



(t, x, y) =



(t, x) 



(t, y) and its Wigner transform



ðx;;tÞ

ð2Þ

iy





t; x þ

y

; x 

y



½35

Then for any t > 0, EW



(t) converges weakly with 

going to zero to a solution F(t) of the kinetic equation

Fðt; x;Þþ r

Fðt; x;Þ

jTð; 

Þj

ðjj

j

ÞðFðt; x;

 Fðt; x;ÞÞd

½36

where T is the amplitude of the scattering operator

associated to the Schro¨ dinger equation with the

short range potential V.

The proof uses several ingredients including

scattering theory with expansion in term of Dyson

series; see Erdo¨ s and Yau (1998).

Semiconductor Modeling

In modern computers, the electronic devices are so

small that the electric current may have no space/time

to reach a thermodynamical equilibrium. Therefore,

this turns out to be a field where the kinetic equations

are the most naturally used. Details of what can be

deduced from a mathematical analysis can be found

in Poupaud (1994). The equations involve the

distribution of electrons f

(x, k, t) and holes

(x, k, t ) and have the following form:

@

ðt; x; kÞþv

ðkÞr

ðt; x; kÞ

h

Uðt; xÞr

ðt; x; kÞ



ðQ

ðf

Þðt; x; kÞþR

ðf

; f

Þðt; x; kÞÞ ½37

@

ðt; x; kÞþv

ðkÞr

ðt; x; kÞ



h

Uðt; xÞr

ðt; x; kÞ



ðf

Þðt; x; kÞþR

ðf

; f

Þðt; x; kÞ½38

Kinetic Equations 205

The variable k ranges over a torus B of R

which, in

physics books, carries the name of Brillouin zone.

The velocities of propagation of electrons and holes

are determined in terms of the energy band by the

formula

e;h

h

e;h

ðkÞ½39

The potential U is determined in terms of the doping

profile C(x), the conductivity 

, and the density of

electrons and holes according to the formula



Uðt; xÞ¼



CðxÞ

jBj

ðt; x; kÞdk



jBj

ðt; x; kÞdk



½40

Finally Q

e,h

and R

e,h

are binary integral operators in

the variable k 2 B which model collisions and

generation–recombination processes. Conce rning

the ‘‘mathematical approach’’ the situation is as

follows.

The relations [39] can be deduced from the high-

frequency analysis of the solution of the Schro¨ dinger

equation

ih@

¼

h

 þ V

h



½41

with V a periodic potential constructed on the dual

lattice of B. The method uses the Bloch decom-

position of the solution and t he Wigner series

(Poupaud 1994). No mathematical derivation of

the collisions operator is currently available. The

situation should be compared to what is said in the

secti on ‘‘Derivation of ki netic equations from the

Schro¨ dinger e quation,’’ but in a much m or e

complicated setting.

On the other hand, the collision operators Q

e,h

and R

e,h

, as given by phenomenological arguments,

have enough good relaxation properties to allow a

rigorous limit of the system [37]–[38] for  going to

zero (Poupaud 1994). This leads to the justification

of the so-called drift–diffusion models and to the

possibility of constructing correctors (with respect to

) and to treating the effect of heterojunctions by

boundary layer analysis.

Some Specific Mathematical Tools

Few proofs were given in the above exposition and

details would not be suitable for a review article.

However, the mathematical approach to kinetic

equations has generated some new tools, and it

may be useful to give the most prominent ones.

The Averaging Lemma

Compactness results appear in spectral theory and in

the construction of solutions of nonlinear equations

(whenever strong convergence is needed for the

limit). Being hyperbolic, the transport operator

v r

propagates singularities along characteristics.

Therefore, at first sight it seems hopeless that one

might obtain any regularizing effect from the free

streaming part of a kinetic model. The key to

obtaining regularizing effects from the transport

operator v r

is to seek those effects not on the

number density itself, but on velocity averages

thereof; in other words, on the macroscopic densities.

Here is the prototype of all velocity averaging

results.

Theorem 4 Let F



be a bounded family in L



). Assume that the family v r



is also bounded

in L

 R

). Then, for each  2 L

), the

family of moments 



(x) defined by





ðxÞ¼



ðx; vÞðvÞ dv

is relatively compact in L

For the proof one starts with the expression



= F



þ v r



takes the Fourier transform with

respect to x of this relation and writes for





() the

expression





ð; vÞðvÞdv

1 þ iv:

½42

Then use the Cauchy–Schwarz inequality to obtain





jðvÞj

1 þjv  j



ð; vÞj

dv ½43

and complete the proof by standard arguments.

The averaging lemma was first observed by

Agoshkov (1984) for abstract results concerning

the regularity of solutions of kinetic equations in

domains with boundary. Independently, it was

rediscovered in the improved form given ab ove by

Golse, Perthame, and Sentis (1985) and used for the

spectral theory in the diffusion approximation. The

extension to L

, p > 1, spaces and to L

(with use of

entropy estimate) were instrumental in proving the

validity of the Rosseland approximation for the

radiative transfer equations and for the proof of

existence by Lions and Di Perna of renormalized

solutions of the Boltzmann equation. A more refined

result needs to be used to establish the incomp res-

sible limit of the solutions of the Boltzmann

equations; see Golse (2005) for details and a

complete list of references.

206 Kinetic Equations