From Hamiltonian particle systems to

Kinetic equations

Mario Pulvirenti

Dipartimento di Matematica Guido CastelnuovoUniversita La Sapienza, P.le Aldo Moro, 2

Roma, Italy

1. Introduction

Many interesting systems in Physics and Applied Sciences are constituted by a large number

of identical components so that they are difficult to analyze from a mathematical point

of view. On the other hand, quite often, we are not interested in a detailed description

of the system but rather in his collective behavior. Therefore it is necessary to look for

all procedures leading to simplified models, retaining all the interesting features of the

original system, cutting away redundant informations. This is exactly the methodology of

the Statistical Mechanics and the Kinetic Theory when dealing with large particle systems.

Here we want to outline the limiting procedures leading, from the microscopic description

of a large particle system (based on the fundamental laws like Newton or Schrodinger

equations), to a the more practical picture dictated by the kinetic theory.

In these notes we present the Boltzmann and the Landau equations, their formal prop-

erties, the scaling limits in which they are expected to hold and some mathematical results.

A particular emphasis is given to the Boltzmann equation which is historically and

conceptually the most interesting equation.

The physical system we are gonig to investigate is constituted by N identical particles

in all the space R3. A state of the system is a sequences z1 . . . zN where zi = (xi, vi) denotes

position and velocity of the i−th particle. The particles interact via a (sufficiently smooth)

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two-body interaction ϕ : R3 → R, so that the equation of motion arexi = vi,

vi = −∑j=1,Nj 6=i

∇ϕ(xi − xj) (1)

We assume, as usual, that ϕ is spherically symmetric so that the force of the particle j

acting on the particle i (that is −∇ϕ(xi − xj)) is directed along xi − xj.We are interested in a situation where N is very large so that we describe the system

from a statistical point of view. More precisely we introduce a probability measure (abso-

lutely continuous with respect to the Lebesgue measure) WN(ZN)dZN on the phase space

of the system R3N ×R3N where ZN = (XN , VN) = (z1 . . . zN) = (xi, vi . . . xN , vN). As usual

the time evolved measure is defined by

WN(ZN ; t) = WN(Φ−t(ZN)) (2)

Here Φt(ZN) denotes the the dynamical flow constructed by solving the equations of

motion, namely Φt(ZN) solves (1) with initial datum ZN .

Since the particles are identical, physically relevant measures are symmetric with re-

spect to the exchange of particles. Consequently we assume:

WN(z1 . . . zi . . . zj . . . zN) = WN(z1 . . . zj . . . zi . . . zN)

for all pairs i 6= j = 1 . . . N .

It is immediate to verify that WN(ZN ; t) solves a partial differential equation, the

Liouville equation

∂tWN(t) = LNW

N(t) (3)

where

LNWN(t) =−

N∑i=1

[vi · ∇xi + Fi · ∇vi ] (4)

−N∑i=1

(vi · ∇xi)−N∑

i,j=1i<j

F (xi − xj) · (∇vi −∇vj)

and where F (x) = −∇ϕ(x) is the force and

Fi = −∑j:j 6=i

∇ϕ(xi − xj)

2

is the force on the particle i exerted by the rest of the system.

Eq.n (3) is equivalent to eq.n (1) and both equations, due to the large number of par-

ticles, are intractable from a practical point of view. To have a more efficient reduced

description, we observe that, being all the particles identical, we would have a good de-

scription of the time evolution whenever we could characterize the time evolution of the

probability distribution of a given particle (say particle 1). To this end we define the

j-particle marginals

fNj (Zj; t) =

∫dzj+1 . . . dzNW

N(Zj, zj+1 . . . zN ; t), j = 1 . . . N. (5)

Obviously fNj (Zj; t) denotes the probability distribution of the first j particles (or of

any given group of j particles) at time t. To find an equation for fNj (Zj; t), we integrate

(3) with respect to dzj+1 . . . dzN . Note that fNN = WN . We find:∫dzj+1 . . . dzN(∂t +

N∑i=1

vi · ∇xi)WN(Zj, zj+1 . . . zN) = (∂t +

j∑i=1

vi · ∇xi)fNj (Zj).

To evaluate the term∫dzj+1 . . . dzN

N∑i=1

Fi · ∇viWN(Zj, zj+1 . . . zN) =

−N∑i=1

N∑k=1,k 6=i

∫dzj+1 . . . dzN∇xϕ(xi − xk) · ∇viW

N(Zj, zj+1 . . . zN),

we readly realize that the terms∑N

i=j+1

∑Nk=1,k 6=i vanish by virtue of the Liouville theorem.

The terms∑j

i=1

∑jk=1,k 6=i reproduce the j-particle Liouville operator.

Finally:

j∑i=1

N∑k=j+1

∫dzj+1 . . . dzN∇xϕ(xi − xk) · ∇viW

N(Zj, zj+1 . . . zN) =

(N − j)j∑i=1

∫dzj+1∇xϕ(xi − xj+1) · ∇vif

Nj+1(Zj, zj+1).

Here we used again the symmetry: the N − j integrations relative to the sum∑N

k=j+1

give the same result.

In conclusion we arrive to the following hierarchy of equations:

∂tfNj (t) = Ljf

Nj (t) + (N − j)Cj+1f

Nj+1, j = 1 . . . N (6)

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where

Cj+1fNj+1(Zj) =

j∑i=1

∫dzj+1∇xϕ(xi − xj+1) · ∇vif

Nj+1(Zj, zj+1) (7)

for j < N and CN+1 = 0. Here we recover the Liouville equation for j = N since fNN = WN .

The physical significance of the hierarchy (6) is transparent. The variation in time of

fNj depends on the interaction of the first j particles among themselves (the j-particle

Liouville operator) and the interaction of the first j particles with the rest of the system.

The name BBKGY for the hierarchy (6) is due to the names of the physicists Bogoli-

ubov, Born, Kirkwood, Green and Yvon, who proposed such set of equations.

Note that the operator Cj+1 maps functions of j+1 particles to functions of j particles,

so that, in order to solve the equation for fNj , we have to know fNj+1. Thus it seems that

there is no advantage to put the problem in this way. In particular to know f1(z1; t) we

have to solve

(∂t + v1 · ∇x1)fN1 (t) = (N − 1)

∫dx2

∫dv2∇xϕ(xi − xj+1) · ∇v1f

N2 (x1, v1, x2, v2), (8)

thus the two-particle distribution should be known. In other words the problem of solving

the BBKGY hierarchy of equations is as hard as solving the original Liouville equation.

However if one could say that

fN2 (x1, v1, x2, v2) = fN1 (x1, v1)fN1 (x2, v2), (9)

namely if the particles could be considered as statistically independent, then eq.n (8) would

be a single nonlinear equation which is much simpler to study than the original problem.

Such a statistical independence is, strictly speaking, false . In facts even though the particle

are distributed independently at time zero, namely

WN(z1 . . . zN) =N∏i=1

f0(zi)

for some initial one-particle distribution f0, then the dynamics creates correlations. Indeed

the position and velocity of particle 1 depend on the positions and velocities of all other

particles. However there are asymptotic situations (when N is large and something else

happens) in which condition (9) can be recovered so that we succed in getting a single

effective equation.

This is the basic strategy of the Kinetic Theory and we shall try to illustrate how it

works for the Boltzmann and Landau equations.

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Kinetic Theory applies to various problems arising from from different areas of sciences.

However the examples of the Boltzmann and Landau equations are paradigmatic, namely

they are the leading examples.

We shall show how these equation are expected to be valid in some asymptotic regimes,

namely in the low-density and weak-coupling limits (for the Boltzmann and Landau equa-

tions respectively). Unfortunately a rigorous mathematical analysis of these limits is still

far to be achieved, although these equations are been established at heuristic level many

years ago. Progresses in this challenging direction are fundamental for the foundation of

Kinetic Theory.

In this notes we shall deal with classical systems, however the same problems can be

formulated for quantum systems as well. We will not review the very few results in this

direction.

2. The Boltzmann equation

2.1 Preliminary considerations

Ludwig Boltzmann established an evolution equation to describe the behavior of a rarefied

gas (1872), starting from the mathematical model of elastic balls and using mechanical and

statistical considerations. The importance of this equation is two-fold. From one side it

provides (as well as the hydrodynamical equations) a reduced description of the microscopic

world. On the other, it is also an important tool for the applications, expecially for dilute

fluids when the hydrodynamical equations fail to hold.

The starting point of the Boltzmann analysis is to renounce to study the gas in terms

of the detailed motion of the molecules which constitute it because of their huge number.

It is rather better to investigate a function f(x, v) which is the probability density of a

given particle, where x and v denote its position and velocity. Actually f(x, v)dxdv is

often confused with the fraction of molecules falling in the cell of the phase space of size

dx dv around x, v. The two concepts are not exactly the same but they are asymptotically

equivalent (when the number of particles is diverging) if a law of large numbers holds.

This is the basic starting point and we spend some more words. We want to consider a

system of N identical particles, whose physical state is given by the sequence of positions

and velocities (z1 . . . zN), where zi = xi, vi being xi and vi the position and velocity of

the i-th particle. We can equally well describe the system in terms of a measure µ in the

5

one-particle phase space, setting

µ(dz) =1

N

N∑i=1

δ(z − zi)dz.

which is called empirical distribution. If z1 . . . zN are identically distributed random vari-

ables with law f = f(z) = f(x, v), then µ is close to f , in a sense specified by the law

of large number, when N diverges. The Boltzmann equation refers to a limiting situation

for which the particle system specified by the state z1 . . . zN can be actually considered

as a large number of copies of independent one-particle system for which µ(dz) and f are

practically the same object.

The Boltzmann equation is the following:

(∂t + v · ∇x)f = Q(f, f) (10)

where Q, the collision operator, is defined by :

Q(f, f) =

∫R3

dv1

∫S2

+

dn (v − v1) · n [f(x, v′)f(x, v′1)− f(x, v)f(x, v1)] (11)

and

v′ =v − n[n · (v − v1)] (12)

v′1 =v1 + n[n · (v − v1)].

Moreover n (the impact parameter) is a unitary vector and S2+ = {n|n · (v − v1) ≥ 0}.

Note that v′, v′1 are the outgoing velocities after a collision of two elastic balls with

incoming velocities v and v1 and centers x and x+ rn, being r the diameter of the spheres

(see fig.1). Obviously the collision takes place if n ·(v−v1) ≥ 0. Eqn.s (12) are consequence

of the conservation of the energy, momentum and angular momentum. Note also that r

does not enter in eqn (10) as a parameter.

A first, very rough justification of eq.n (10) is the following. If the particles would

not collide, then the evolutin equation for the probability distribution of a single particle

should be

(∂t + v · ∇x)f = 0.

However, due to the collisions, we have an equation like

(∂t + v · ∇x)f = G− L = Q,

6

fig.1

v

v

v

v

1

‘

1‘

n

where G, the gain term, describes how the fraction of particles in the cell dxdv of the

phase-space (that is fdxdv) increases, due to the collisions of other particles in different

cells. For the same reason L describes the loss of the quantity fdxdv due to the fact that

a particle in dxdv can collide and hence disappear from that cell. The explicit form of G

and L given by eq.n (11) will be discussed later on.

As fundamental features of eq.n (10) we have the formal conservation in time of the

following five quantities ∫dx

∫dvf(x, v; t)vα (13)

with α = 0, 1, 2, expressing conservation of the probability, momentum and energy.

From now on we shall set∫

=∫

R3 for notational simplicity.

Boltzmann introduced the (kinetic) entropy defined by

H(f) =

∫dx

∫dvf log f(x, v) (14)

and proved the famous H-theorem asserting the decreasing of H(f(t)) along the solutions

to eq.n (10).

Finally, in case of bounded domains or homogeneous solutions (f = f(v; t) is indepen-

dent of x), the distribution defined for some β > 0, ρ > 0, and u ∈ R3, by:

M(x, v) =ρ

(2πβ

)3/2e−

β2|v−u|2 , (15)

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called Maxwellian distribution, is stationary for the evolution given by eq,n (10). In addi-

tion M minimizes H among all distributions with given total mass ρ, given mean velocity

u and mean energy. The parameter β is interpreted as the inverse temperature.

In conclusion Boltzmann was able to introduce an evolutionary equation with the re-

markable properties of expressing mass, momentum, energy conservations, but also the

trend to the thermal equilibrium. In other words he tried to conciliate the Newton laws

with the second principle of thermodynamics.

2.2 First consequences

The Boltzmann equation provoked a debate involving Loschmidt, Zermelo and Poincare

who outlined inconsistencies between the irreversibility of the equation and the reversible

character of the Hamiltonian dynamics. Boltzmann argued the statistical nature of his

equation and his answer to the irreversibility paradox was that “most” of the configu-

rations behave as expected by the thermodynamical laws. However he did not have the

probabilistic tools for formulating in a precise way the statements of which he had a precise

intuition.

H. Grad (1949) stated clearly the limit N → ∞, r → 0, Nr2 → const, where N is

the number of particles and r is the diameter of the molecules, in which the Boltzmann

equation is expected to hold. This limit is usually called the Boltzmann-Grad limit .

The problem of a rigorous derivation of the Boltzmann equation was an open and

challenging problem for a long time. O. E. Lanford (1975) showed that, although for

a very short time, the Boltzmann equation can be derived starting from the mechanical

model of the hard-sphere system.

Now we show that the properties we established so far are actually consequence of the

equation.

Note first that in the definition of the collision operator, the spatial variable x enters

as a parameter, so that, in the following algebraic manipulation we skip this dependence.

Let ϕ = ϕ(v) a given test function, consider the scalar product in L2(v):

(ϕ,Q(f, f))) =

∫dv

∫dv1

∫S2

+

dn B(v − v1;n) ϕ[f ′f ′1 − ff1]. (16)

Here we use the standard notation f = f(v), f ′ = f(v′), f1 = f(v1), f ′1 = f(v′1). Using the

symmetry v → v1 and the fact that the Jacobian of the transformation (12) is unitary, we

readly arrive to

(ϕ,Q(f, f))) =1

2

∫dv

∫dv1

∫S2

+

dn B(v − v1;n) ff1[ϕ′ + ϕ′1 − ϕ− ϕ1]. (17)

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Therefore the conservation of mass, momentum and energy follows from the fact that

(ϕ,Q(f, f))) = 0 (18)

whenever ϕ = 1, v, v2. Finally the H-theorem is consequence of the identity

H(t) = (log f,Q(f, f))), (19)

where the scalar product is with respect the x, v variables. By using (17) with ϕ = log f ,

we arrive to

(log f,Q(f, f))) =1

4

∫dx

∫dv

∫dv1

∫S2

+

dn B(v − v1;n) [ff1 − f ′f ′1] logf ′f ′1ff1

. (20)

The right hand side of (20) is manifestly negative, so that a formal proof of the H-theorem

is given.

Finally we establish the Gibbs principle which states that the Maxwellian (15) mini-

mizes H among all probability densities with prescribed mean velocity and energy. This is

a trivial exercise of calculus of variation which we leave to the reader.

2.3 The hard-sphere system

Our starting point is a system of N identical hard spheres of diameter ε of unitary

mass, interacting by means of the collision law (12) . We denote by ε the diameter of the

particles in view of the Boltzmann-Grad limit (ε → 0, N → ∞, Nε2 → cost) we want to

discuss. For the moment, however, N and ε are fixed.

We denote by ZN = (z1, . . . , zN) = (XN , VN) = (x1, . . . , xN ; v1, . . . , vN) a state of the

system, where zi = (xi, vi) ∈ R6. The phase space ΓN of the system is the subset of the

phase space (R6)N fulfilling the hard-core condition, namely

|xi − xj| ≥ ε for i 6= j.

The dynamical flow is defined as the free flow (namely ZN(t) = (x1 + v1t, . . . , xN +

vN t; v1, . . . , vN)) up to the first impact time namely when |xi− xj| = ε. Then an instanta-

neous collision takes place, according to the law (12) and the flow goes on up to the next

collision instant.

We denote by ZN → Φt(ZN) the dynamical flow constructed in this way.

It is not obvious that the such a flow makes sense. Indeed some pathologies can occur,

namely a triple collision or an accumulation of collision instants to some finite value. After

those situations we do not know how to go on with the flow. Fortunately it is possible

9

to show that the set of initial configurations leading to those pathologies has a vanishing

Lebesgue measure, [10] and references quoted therein.

We do not discuss further this point and we shall use ZN → Φt(ZN) uniquely definded

by the collision law, for almost all ZN .

Given a probability measure with density WN0 on ΓεN , thanks to the invariance of the

Lebesgue measure under above evolution, we define, as usual, the time evolved measure as

the one with density

WN(ZN ; t) = WN0 (Φ−tZN). (21)

Through the present analysis we will consider that the probability distribution WN0 is

initially (and hence at any positive time) symmetric in the exchange of the particles. We

are interested in the evolution of the one-particle distribution fN1 (x, v; t).

In general the j-particle distributions fNj of a measureWN (sometimes called marginals)

are defined as:

fNj (z1, . . . zj; t) =

∫dzj+1 . . . dzN WN(z1, . . . zN ; t). (22)

They denote the probability density of the first group of j-particles or of any other

group of j tagged particles, due to the symmetry of the measure WN .

For ZN ∈ ΓN we have

∂

∂tWN(ZN , t) = −

N∑j=1

vj · ∇xjWN(ZN , t).

Integrating on the last N − 1 variables, we have:

∂

∂tf1(x, v; t) = −

N∑j=1

∫Γ(x1)

vj · ∇xjWN(x1, v1 . . . xN , vN)dx2, dv2 . . . dxN , dvN ,

where

Γ(x1) = {x2, v2 . . . xN , vN ||x1 − xs| > ε, s = 2 . . . N}.

The term j = 1 in the above sum can be handled according to the formula

v1 · ∇x1f1(x1, v1) =v1 · ∇x1

∫Γ(x1)

WN(x1, v1 . . . xN , vN)dx2, dv2 . . . dxN , dvN = (23)∫Γ(x1)

v1 · ∇x1WNdx2, dv2 . . . dxN , dvN+

N∑s=2

∫∂Γs(x1)

v1 · nsWNdx2, dv2 . . . dσs, dvs . . . dxN , dvN ,

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where where

∂Γs(x1) = {x2, v2 . . . xN , vN ||x1 − xr| > ε, r = 2 . . . N, r 6= s, |x1 − xs| = ε}.

The last is a boundary term arises from the fact that the derivative with respect to x1

involves also the domain Γ(x1). σs is the element of the surface {xs||xs − x1| = ε} and ns

denotes the outward normal. Using the symmetry of WN we write the last term as

(N−1)

∫dv2

∫dσ2 v1·n2 f2(x1, v1, x2, v2) = (N−1)ε2

∫dv2

∫S2

dn v1·n f2(x1, v1, x1+εn, v2).

Finally we can use the div lemma to prove that

N∑j=2

∫Γ(x1)

vj · ∇xjWN(x1, v1 . . . xN , vN)dx2, dv2 . . . dxN , dvN (24)

N∑j=2

∫∂Γj(x1)

vj · njWNdx2, dv2 . . . dσjdvj . . . dxN , dvN = (25)

(N − 1)

∫dv2

∫dσ2v2 · n2f2(x1, v1, x2, v2) =

(N − 1)ε2

∫dv2

∫dnv2 · nf2(x1, v1, x1 + εn, v2).

In conclusion we obtain:

(∂

∂t+ v1 · ∇x1)f1(x1, v1; t) = (N − 1)ε2

∫dv2

∫dnf2(x1, v1, x1 + nε, v2)(v2 − v) · n. (26)

However eq.n (26) is not very useful because the time derivative of f1 is expressed

in term of another object, namely f2. An evolution equation for f2 will imply f3, the

joint distribution of three particles and so on up to arrive to the total particle number N .

Here the basic Boltzmann’s main assumption enters, namely that two given particles are

uncorrelated if the gas is rarefied:

f2(x, v, x2, v2) = f1(x, v)f1(x2, v2) (27)

Condition (27), called propagation of chaos, seems to be contradictory at a first sight: if

two particles collide, correlations are created. Even though we could assume eq.n (27) at

some time, if the test particle (say 1) collides with the particle 2 such an equation cannot

be satisfied anymore after the collision.

Before discussing the propagation of chaos hypothesis, we first analyze the size of

the collision operator, namely the right hand side of (26). We remark that, in practical

11

situations for a rarefied gas, the combination Nε3 ≈ 10−4cm3 (that is the volume occupied

by the particles) is very small, while Nε2 = O(1). This implies that the interaction term

in (26) is O(1). Therefore, since we are dealing with a very large number of particles, we

are tempted to perform the limit N →∞ and ε→ 0 in such a way that ε2 = O(N−1), that

is the Boltzmann-Grad limit. As a consequence the probability that two tagged particles

collide (which is of the order of the surface of a ball, that is O(ε2)) is negligible. However

the probability that a given particle performs a collision with any one of the remaining

N − 1 particles (which is O(Nε2) = O(1)), is not negligible. Therefore condition (27)

is referring to two preselected particles (say particle 1 and particle 2) so that it is not

unreasonable to conceive that it holds in the limiting situation in which we are working.

However we cannot insert (27) in (26) because this latter equation refer both to instants

before and after the collision and, if we know that a collision took place, we certainly cannot

invoke eq.n (27). Hence it is more convenient to assume eqn (26) before the collision, namely

when (v2 − v1) · n > 0. Therefore we split the integral in the right hand side of (26) into

two terms

G = (N − 1)ε2

∫dv2

∫(v2−v1)·n>0

dnf2(x, v, x+ nε, v2)(v2 − v1) · n, (28)

L = (N − 1)ε2

∫dv2

∫(v2−v1)·n<0

dnf2(x, v, x+ nε, v2)(v2 − v1) · n, (29)

Here G and L stand for gain and loss term.

We now make use of the continuity property

f2(x, v, x+ nr, v2) = f2(x, v′, x+ nr, v′2) (30)

where the pair v′, v′2 is pre-collisional. On f2 expressed before the collision we can reason-

ably apply condition (27) obtaining:

G− L = (N − 1)r2

∫dv2

∫S2

+

dn(v − v2) · n (31)

×[f(x, v′)f(x− nr, v′2)− f(x, v)f(x+ nr, v2)]

after a change n → −n in the Gain term, using the notation S2+ for the hemisphere

{n| = (v2 − v) · n ≥ 0}. This transform the pair v′, v′2 from a pre-collisional to a post-

collisional pair.

Finally, in the limit N →∞, r → 0, Nr2 = λ−1 we find:

(∂t + v · ∇x)f =

12

λ−1

∫dv2

∫S+

dn (v − v2) · n [f(x, v′)f(x, v′2)− f(x, v)f(x, v2)]. (32)

The parameter λ, called mean free path, represents, roughly speaking, the typical length a

particle can cover without undergoing any collision. In eqn (10) we just choosed λ = 1.

The Boltzmann equation has a statistical nature and it is not equivalent to the Hamil-

tonian dynamics from which it has been derived. Indeed the H-theorem shows that such

an equation is not reversible in time as the law of mechanics.

This concludes the heuristic preliminary analysis of the Boltzmann equation. We cer-

tainly know that the above arguments are delicate and require a more rigorous and deeper

analysis. If we want that the Boltzmann equation is not a phenomenological model, derived

by assumptions ad hoc and justified by its practical relevance, but rather a consequence of

a mechanical model, we must derive it rigorously. In particular the propagation of chaos

should be not an hypothesis but the statement of a theorem.

2.4 Hierarchies

We are interested in the behavior of the system in the limit N → ∞, ε → 0 fixing the

combination ε2N = 1 (1 is choosen just for simplicity) according to the Boltzmaqnn-Grad

limit. Therefore we have a single parameter ε and we want to study the asymptotics for

ε→ 0.

Of course the probability distribution W ε(t) is not expected to converge to something

because it is defined on R6N which is changing with the sequences. Thus we introduce the

marginal distributions for which the convergence problem makes at least sense. We set:

fNj (z1, . . . zj; t) =

∫dzj+1 . . . dzN WN(z1, . . . zN ; t). (33)

and want to exploit the limit N →∞ (or ε→ 0) for any fixed j = 1, 2 . . . .

We have formally derived an equation for fN1 (in terms of fN2 )

There is an evolution equation for the set of distribution functions {fNj }Nj=1 (in the

same spirit of the well known BBKGY hierarchy) that is

(∂t + Lεj)f

Nj = (N − j)ε2Cε

j+1fNj+1, j = 1 . . . N (34)

where Lεj is the generator of the dynamics of j hard-cores of diameter ε and:

Cεj+1f

Nj+1(x1, v1, . . . xj, vj) =

j∑k=1

∫dn

∫dvj+1n · (vk − vj+1) (35)

fNj+1(x1, v1, . . . xj, vj, . . . , xk − εn, vj+1)

13

where n is the unit vector.

Here we have denoted Lεj the Liouville operator relative to the dynamics of the j-particle

system (see Remark below) . Finally we set fNj = 0 if j > N and hence, for j = N we

have nothing else than the Liouville equation.

Remark.

What we mean by the generator of the dynamics of j hard-cores of diameter ε ? Fixed

j such that 2 ≤ j ≤ N , consider the j-particle flow Φt. This is defined on Γεj which is

the j-particle phase space with the hard-core condition. Suppose that the smooth function

F = F (Zj) fulfills the property

F (Zj) = F (Z ′j); DF (Zj) = DF (Z ′j) (36)

for any Zj in the boundary {Zj||xi − xk| = ε for some i 6= k}. Here we assume that

Zj has the the pair vi, vk incoming and Z ′j differs from Zj only for replacing vi, vk by the

outgoing pair v′i, v′k. Finally D denotes any first derivative. Defining

U ε(t)F (Zj) = F (Φ−t(Zj)) (37)

then U ε(t)F is differentiable and

d

dtU ε(t)F (Zj) =

d

dtF (Φ−t(Zj)) = Lε

jF (Φ−t(Zj)).

This means that Lεj =

∑ji=1 vi · ∇xi plus boundary conditions (36) at ∂Γεj . ER

Remark.

Eq.n (34) is noting else than a rephrasing of the Liouville equation

(∂t + LεN)WN = 0. (38)

Solving (38) from (34) we recover fNN−1(t) and so on. Strictly speaking eq.n (34) is not an

equation but a sequence of identities starting from (38) which we have to solve. ER

There are many aspects of eq.n (34) which have to be clarified and which will be

discussed in Appendix. We observe that the first eq.n of the hierarchy has been heuristically

obtained when deducing the Boltzmann equation (see eq.n (26)).

Moreover note that the factor in front of the collision operator Cεj+1 is O(1) so that we

can hope to have a limit.

The physical meaning of eq.ns (34) is the same as for the usual BBKGY hierarchy :

the variation in time of the j-particle marginal f εj is due to two effects. The first is that

it changes because the interaction of the first group of j particles (the action of Lεj ). The

14

second is due to interaction of this group with the rest N − j particles. This part of the

interaction is O(1) while the action of Lεj should be negligible in the limit we are analyzing.

Indeed it is obvious that, for any fixed j

limε→0

U εj (t)F = S(t)F

where S is the free flow

S(t)F (Xj, Vj) = F (Xj − V t;Vj).

and the above limit holds almost everywhere.

From now on we shall use ε as a unique parameter to be sent to zero, being N = ε−2

authomatically rescaled. We shall represent the time evolved marginals f εj (t) = fNj (t) by

means of a perturbative expansion (this is just the iteration of the Duhamel formula):

f εj (t) =

N−j∑m=0

αεm(j)

∫ t

0

dt1

∫ t1

0

dt2 . . .

∫ tm−1

0

dtm (39)

U ε(t− t1)Cεj+1 . . . U

ε(tm−1 − tm)Cεj+mU

ε(tm)f ε0,m+j.

where f ε0,j are defined by eq.n (33) in terms of the initial probability distribution W ε0 (on

which we have still to do suitable hypotheses). Finally:

αεm(j) = ε2m(N − j)(N − j − 1) . . . (N − j −m+ 1). (40)

Obviously αεm(j) = O(1).

Note that eq.n (39) is an identity which express f εj in terms of a finite sum of operators

acting on the initial sequence f ε0,j. Moreover f εj is well defined a.e. by the existence of the

flow Φt. On the other hand the right hand side of (39) makes sense and (39) establishes an

identity between L1 functions for all t ≥ 0. Remind that for j = N we have the identity

f εN(t) = WN(t).

2.5 The tree expansion

The j-particle marginals can be represented in terms of a tree expansion which will be

useful in the sequel and allow us a better control of the expansion in the form (39). We set

Cεj+1 =

j∑k=1

Cεk,j+1 (41)

Cεk,j+1 = Cε+

k,j+1 − Cε−k,j+1 (42)

15

and

Cε+k,j+1gj+1(x1, . . . , xj; v1, . . . , vj) =

∫dvj+1

∫S2

+

dωω · (vk − vj+1)

[gj+1(x1, . . . , xj, xk − εω; v1, . . . , v′k . . . , v

′j+1), (43)

Cε−k,j+1gj+1(x1, . . . , xj; v1, . . . , vj) =

∫dvj+1

∫S2

+

dωω · (vk − vj+1)

gj+1(x1, . . . , xj, xk + εω; v1, . . . , vk, . . . , vj+1)]. (44)

Here we did the same algebraic manipulations use for getting (31).

Therefore

f εj (t) =

N−j∑n=0

∑σ1,...,σnσi=±1

∑k1,...,kn

′(−1)|σn|∫ t

0

dt1

∫ t1

0

dt2 . . .

∫ tn−1

0

dtn (45)

U ε(t− t1)Cεσ1k1,j+1 . . . U

ε(tn−1 − tn)Cεσnkn,j+n

U ε(tn)f ε0,n+j.

Here

σn = (σ1, . . . , σn), σi = ± |σn| =∑j

σj1

and ∑k1,...,kn

′ =

j∑k1=1

j+1∑k2=1

· · ·j+n−1∑kn=1

It is convenient to represent the expansion (45) in terms of integrals of the initial datum. To

make more explicit this dependence, we introduce a n−collision, j−particle tree, denoted

by G(j, n), as a collection of integers k1 . . . kn such that:

k1 ∈ Ij, k2 ∈ Ij+1, . . . kn ∈ Ij+n

where Is = {1, 2, . . . s}.The name tree is justified by the fact that it has a natural graphical representation.

For instance G(2, 5) given by 1, 2, 1, 3, 2 can represented as in Figure.

Each branch (say j+ `) represents a new particle (with incoming or outgoing velocities

according to σ` = −1 or σ` = 1 respectively) created at time t` by a previous particle

(branch) k` = 1, . . . j + `− 1.

16

The set of all such trees is denoted by G(j, n). In the following we shall write∑k1...kn

′ =∑

G(j,n)∈G(j,n)

.

Note that the number of terms is j(j + 1) . . . (j + n− 1).

At time s ∈ (0, t) we denote by ζε(s) = (ξε(s), ηε(s)) the vector of the positions and

velocities of the particles created up to the time s. If s ∈ (tr, tr+1) we have just j + r

particles whose positions and velocities are:

ξε(s) = (ξε1(s), . . . , ξεj+r(s))

and

ηε(s) = (ηε1(s), . . . , ηεj+r(s))

respectively.

The particle j + r is created at time tr by particle i in the position

ξεj+r(tr) = ξεi (tr)− σrωrε

with velocity

ηj+r(t−r ) = vj+r + ωr · (ηεi (t+r )− vj+r)ωr(

σr + 1

2)

Note that ηj+r(t−r ) = vj+r if σr = −1 and, accordingly, ωr · (ηεi (t+r )− vj+r) ≥ 0. In this case

ηεi (t+r ) = ηεi (t

−r ). If σr = 1 the pair (ηεi (t

+r ), vj+r)) is post-collisional. Then also the velocity

of the direct progenitor i changes according to the formula:

ηi(t−r ) = ηi(t

+r )− ωr · (ηεi (tr)− vj+r)ωr.

The two situations are illustrated in figure.

Once established the initial position and velocities of the new created particles, the

(backward) flow is Φ−t up to the next creation in which the procedure is repeated.

17

! =-1 ! =1

"r "

r

v

j+r

vi

#= v

i

+

vi

+

v

j+r

v

j+r

#

vi

#

fig.2

It is very important to note that between two creation instants tr, tr+1 the particles

can collide when at distance ε. These collisions outside the creation times are called

recollisions, because each particle of the group j + r already interacted (at the creation

time) with another particle of that group. We also notice that recollisions are unlikely, but

they are the main responsible of the different behavior of the particle dynamics from the

Boltzmann evolution. This will be further clarified later on.

The configuration ζε(s) depends on G(j, n), on the creation times (t1, . . . tn), on the

sequence σn and on the velocities of the created particles vj+1, . . . , vj+n. We do not explicit

such a dependence for sake of notational simplicity.

Next we set

tn = (t1, . . . , tn),

ωn = (ω1, . . . , ωn)

Vj,n = (vj+1, . . . , vj+n)

σn = (σ1, . . . , σn), σj = ±1

and define

dΛ(tn, ωn,Vj,n) = χ({t1 > t2 · · · > tn})dt1 . . . dtndω1 . . . dωndvj+1 . . . dvj+n.

With these definitions we rewrite (45) as

f εj (Zj; t) =

N−j∑n=0

αεn(j)∑

G(j,n)∈G(j,n)

∑σn

(−1)|σn|

∫dΛ(tn, ωn,Vj,n)

n∏i=1

B(ωi; ηεki

(ti)− vj+i) f ε0,j+n(ζε(0)), (46)

18

where B(ωi; ηεki

(ti) − vj+i) = ωi · (ηεki(ti) − vj+i)) and ki is the progenitor of the particle

j + i.

We now treat the solution to the Boltzmann equation in the same manner. Let f be a

solution to eq.n (10). Then consider the products

fj(Zj; t) = f(t)⊗j(Zj) (47)

Then it is easy to show that fj(t) solves the hierarchy of equations

(∂t +

j∑k=1

vk · ∇xk)fj = Cj+1fj+1, (48)

where

Cj+1 =

j∑k=1

Ck,j+1 (49)

Ck,j+1 = C+k,j+1 − C

−k,j+1 (50)

and

C+k,j+1gj+1(x1, . . . , xj; v1, . . . , vj) =

∫dvj+1

∫S2

+

dωω · (vk − vj+1)

[gj+1(x1, . . . , xj, xk; v1, . . . , v′k . . . , v

′j+1), (51)

C−k,j+1gj+1(x1, . . . , xj; v1, . . . , vj) =

∫dvj+1

∫S2

+

dωω · (vk − vj+1)

gj+1(x1, . . . , xj, xk; v1, . . . , vk, . . . , vj+1)]. (52)

In other words we have the same structure as before just putting ε = 0. The above

infinite hierarchy of equations (which does not expresses nothing else than the Boltzmann

equation) is called the Boltzmann hierarchy.

By perturbing around the free flow, we express the solution to (48) by means of a series

expansion:

fj(t) =∞∑n=0

∫ t

0

dt1

∫ t1

0

dt2 . . .

∫ tn−1

0

dtn

S(t− t1)Cj+1 . . . S(tn−1 − tn)Cj+nS(tn)f0,n+j. (53)

19

where

f0,n+j = (f0)⊗(n+j)

and S(t)gj is the free flow.

We can do the same tree expansion as before readily arriving to the following expression

fj(Zj; t) =∞∑n=0

∑G(j,n)∈G(j,n)

∑σn

(−1)|σn|

∫dΛ(tn, ωn,Vj,n)

n∏i=1

B(ωi; ηki(ti)− vj+i) f0,j+n(ζ(0)), (54)

where B(ωi; ηki(ti) − vj+i) = ωi · (ηki(ti) − vj+i)) and ki is the progenitor of the particle

j + i.

Here the backward flow ζ(s) is constructed as before with the difference that the new

particles are created exactly in the same place of their progenitor. Obviously we do not have

recollisions (particles are points). In other words everything goes as before just putting

formally ε = 0.

2.6 Rigorous derivation of the Boltzmann equation for hard spheres

There are various differences between the two expansions (46) and (54) which we want

to outline. Given a tree G(j, n), the sequences of the creation times (t1, . . . tn), σn and the

velocities of the created particles vj+1, . . . , vj+n, the difference between ζ(0)) and ζε(0)) is

due to different initial positions of the new created particles and, possibly, to the occurrence

of recollisions. Note also that, while the sum in (46) is finite, that in (54) is infinite.

Moreover αεn(j) in (46) is replaced by 1 in (54) . Finally the initial data also differs.

Up to now we did not make any hypothesis on the initial data. We assume that f0 is

a continuous, bounded probability density, satisfying the bound

f0(x, v) ≤ Ce−βv2

(55)

We cannot choose, as initial condition for the hard-sphere hierarchy, the same as for

the Boltzmann hierarchy, namely

fj(Zj; 0) = f⊗j0 (Zj) (56)

because the above distributions do not satisfy the hard-core condition which creates an

intrinsic correlation.

20

We assume that the initial data for (46) satisfy

|f ε0,j(Xj, Vj)| ≤ zje−βV2j (57)

for some z > 0. Here Xj = (x1, . . . xj),Vj = (v1, . . . vj) and V 2j =

∑ji=1 v

2i .

Moreover we require that

limε→0

f ε0,j = f⊗j0 (58)

and the above convergence holds, for all j, uniformly on compact sets outside the manifold

{Zj|xi = xs, for some i 6= s}.

By means of an example we could choose, as initial condition, the measure W ε0 which

has the minimal correlations given by the hard-core condition, namely:

W ε0 (ZN) = Z−1

N

N∏i=1

f0(zi)∏i 6=j

χ({|xi − xj| > ε}) (59)

where ZN is the normalization factor

ZN =

∫dZN

∏1≤i<j≤N

χi,j

N∏i=1

gε0(zi).

Here we use the notation χ({. . . }) for the characteristic function of the event {. . . } and

set χi,j = χ({|xi − xj| > ε}).It is not obvious that hypothesis (58) is indeed satisfied. We shall show this in Appendix.

Our goal is to prove:

Theorem 1 (Lanford 75)

Under hypotheses (57), (58) and (56), there exists t0 > 0 such that, for t ≤ t0 and for

all j = 1, 2, . . .

limε→0

f εj (t) = fj(t) a.e. (60)

Moreover

fj(t) = f(t)⊗j a.e., (61)

where f(t) solves the Boltzmann equation.

The proof of the above Theorem consists into two steps.

1) The series (actually it is a sum) (39) and the series (53) are bounded by a series of

positive terms, independent of ε, converging for a short time.

2) Each term of the series (39) converges to the corresponding term of (53).

21

Proof of step 1.

We introduce the norm, for β > 0

‖fj‖β = supXj ,Vj

eβV2j |fj(Xj, Vj)|. (62)

We have the following bound, for β′ > β,

‖Cεj+1fj+1‖β =

C

(β′)3/2

(√ j

β′ − β+

j√β′)‖fj+1‖β′ . (63)

Indeed:

eβV2j |Cε+

j+1fj+1(Xj, Vj)| ≤ 4π‖fj+1‖β′j∑

k=1

∫dvj+1(|vk|+ |vj+1|)e−β

′v2j+1e−(β′−β)V 2j . (64)

Here we used that

(v′k)2 + (v′j+1)2 = v2

k + v2j+1.

Moreover, by the Cauchy-Schwarz inequality

j∑k=1

(|vk|+ |vj+1|) ≤√j|Vj|+ j|vj+1|. (65)

This term, as well the integral in dvj+1, can be controlled by the exponentials. Then (64)

follows easily.

Note that we have the same estimate for ‖Cε−j+1fj+1‖β and for ‖Cj+1fj+1‖β.

Next we observe that the energy conservation, tells us that

‖U ε(t)fj+1‖β = ‖fj+1‖β, ‖S(t)fj+1‖β = ‖fj+1‖β. (66)

As a consequence we have

‖U ε(t− t1)Cεj+1 . . . U

ε(tn−1 − tn)Cεj+nU

ε(tn)f ε0,n+j‖β/2 ≤ (67)

C(β)n(√

jn+ j)(√

(j + 1)n+ (j + 1)). . .(√

(j + n− 1)n+ j + n− 1)‖f ε0,j+n‖β.

The above inequality is obtained by partitioning the interval (β/2, β) into the subintervals

(βk, βk+1) with βk+1 = βk + β/2n, k = 0 . . . n − 1, β0 = β/2. Then we estimate the term

‖Cj+k(. . . )‖βk in terms of the norm ‖(. . . )‖βk+1by using (63). The result is (67).

Since j is fixed, we consider n > j. Then, using (57)

r.h.s. of (67) ≤ C(β)n2nzj+n√jn√

(j + 1)n . . .√

(j + n− 1)n (68)

≤ zj(2C(β)z)nnn/2√j(j + 1) . . . (j + n− 1) (69)

≤ (√

2z)j(√

2C(β)z)nnn/2√n!

22

For the last step we used the following inequality

j(j + 1) . . . (j + n− 1) ≤ n!

(j + n− 1

j − 1

)≤ n! 2j+n−1. (70)

Finally observe that ∫ t

0

dt1

∫ t1

0

dt2 . . .

∫ tn−1

0

dtn ≤tn

n!(71)

and αεm(j) ≤ 1. Thus the generic term of (39) is bounded, for n > j, by

(√

2z)j(√

2C(β)zt)nnn/2√n!. (72)

Finally by using the Stirling formula for which nn ≤ n!Cn, we conclude step 1 because

the series (39) is bounded by a geometric series which is converging for t < t0. Note that

t0 is explicitely computable in terms of the constants in the game. Obviously the same

arguments apply to (53). �

Proof of step 2.

For a fixed j ≥ 1, we set

Tεn(j) =∑

G(j,n)∈G(j,n)

∑σn

(−1)|σn|∫dΛ(tn, ωn,Vj,n)Bε

n fε0,j+n(ζε(0)). (73)

Here Vj,n = (v2, . . . , vj+n).

The analogous expression for the Boltzmann hierarchy is

Tn(j) =∑

G(j,n)∈G(j,n)

∑σn

(−1)|σn|∫dΛ(tn, ωn,Vj,n)Bn f0,j+n(ζ(0)). (74)

Here we set:

Bεn =

n∏i=1

B(ωi; ηεki

(ti)− vj+i); Bn =n∏i=1

B(ωi; ηki(ti)− vj+i)

To show step 2 we have to prove that, for almost all Zj = (Xj, Vj) and all t

αεn(j)Tεn → Tn (75)

as ε → 0. Observing that αεn(j) → 1, by using the dominated convergence theorem, it is

enough to show that, for all z and t:

Bεn f

ε0,n+1(ζε(0))→ Bn f0,n+1(ζ(0)) (76)

23

for any tree G(j, n) , any choice of σn and for almost all Zj, (tn, ωn,Vj,n).

Actually the set Zj must exclude the zero measure manifold

M = {Zj|xi − vis 6= xk − vks, for all i 6= k and s > 0}. (77)

By construction ζε(0) → ζ(0). In facts the recollisions take place for a vanishing

measure set of (tn, ωn,Vj,n) whenever Zj /∈ M. This fact is completely obvious and it is

enough for our purposes.

As a consequence Bεn → Bn. Thus we have to prove that

f ε0,n+1(ζε(0))− f0,n+1(ζ(0)) = f ε0,n+1(ζε(0))− f0,n+1(ζε(0)) (78)

+ f0,n+1(ζε(0))− f0,n+1(ζ(0))

is vanishing as ε→ 0. By using the continuity of f0 and the convergence ζε(0)→ ζ(0) the

second term of the right hand side of (78) is vanishing. The first one is also vanishing by

(58). �

Up to now we have proven that, for all t < t0 , all j and a.a. Zj, the expansion (46)

converges, as ε→ 0 to the corresponding expansion (54). On the other hand, if the initial

condition f0,j factorizes, it is immediate to verify that

fj(t) = f1(t)⊗j. (79)

For instance see the the tree described in figure . If the two branches generated by perticle

1 and 2 do not interact (this is the case when we do not have recollisions), the contributions

to f2(z1, z2, t) is just the product of the contributions given by each branch thought as a

single tree and this is pretty obvious. Note that the tree expansion is crucial to understand

in a direct way the meaning of the propagation of chaos.

Finally using the above factorizetion property it follows that f1 (up to now defined in

terms of the expansion (54) for j = 1) is also a solution of the Boltzmann equation in the

mild form:

f1(t) = S(t)f0 +

∫ t

0

dτS(t− τ)Q(f1(τ), f1(τ)) (80)

where Q is defined in (11).

2.7 Estimate of the error

In this section we want to go beyond the Lanford’s theorem by giving an explicit

estimete of the error

Ej(Zj, t) = f εj (Zjt)− fj(Zj, t)

24

For this purpose it is convenient to introduce the following equation, called Enskog equa-

tion:

(∂t + v · ∇x)gε(x, v; t) = Qε(g

ε, gε)(x, v; t) (81)

where

Qε(gε, gε)(x, v; t) =

∫dv1

∫S2

+

dωω · (v − v∗) (82)

[gε(x, v′)gε(x− εω, v′∗)− gε(x, v)gε(x+ εω, v∗))].

For gε we assume the same iniyial datum than for the Boltzmann equation: gε(0) = f0

We do not give any physical significance to the Enskog evolution (81): it has been

introduced as useful technical bridge between the hard-sphere dynamics and the Boltzmann

equation. Of course gε(t) and f(t) are close when ε is small. In fact we have the following

Proposition 2 Let gε and f be solutions to the initial value problem (81) and (10)

respectively, both with initial datum f0. Assume f0 ∈ C1 in addition to hypotheses (57),

(58) and (56).

Then there exist t0 > 0 such that, for t < t0,

gε(z, t)− f(z, t) = O(ε). (83)

The proof of the existence of gε and the above convergence are straightforward and

easily achieved by using the the tree expansions technique. For this reason we omit it.

It is natural to introduce gεj (t) = (gε(t))⊗j. Clearly gεj (t) solves the hierarchy of equa-

tions, called the Enskog hierarchy,

(∂t +

j∑k=1

vk · ∇xk)gεj = Cε

j+1gεj+1, (84)

where the collision operator Cεj+1 is given by (49),(50),(51) and (52).

In other words we have the same structure of the BBKGY hierarchy, neglecting the

recollosions.

Again we have

gεj (t) =∞∑n=0

∫ t

0

dt1

∫ t1

0

dt2 . . .

∫ tn−1

0

dtn

S(t− t1)Cεj+1 . . . S(tn−1 − tn)Cε

j+nS(tn)f0,n+j. (85)

25

where

f0,n+j = (f0)⊗(n+j)

and S(t)gj is the free flow.

Moreover we can do the same tree expansion as before, readily arriving to the following

expression

gεj (Zj; t) =∞∑n=0

∑G(j,n)∈G(j,n)

∑σn

(−1)|σn|

∫dΛ(tn, ωn,Vj,n)

n∏i=1

B(ωi; ηki(ti)− vj+i) f0,j+n(ζε(0)), (86)

where, as before, B(ωi; ηki(ti)− vj+i) = ωi · (ηki(ti)− vj+i)) and ki is the progenitor of the

particle j + i.

Now the backward flow ζε(s) is constructed as ζε(s) regarding the positions at the

creation time of the new particles, but the recollisions are ignored.

Next, by virtue of the above Proposition, we are reduced in controlling the difference

between f εj and gεj .

Setting

Tεn(j) =∑

G(j,n)∈G(j,n)

∑σn

(−1)|σn|∫dΛ(tn, ωn,Vj,n)Bε

n fε0,j+n(ζ

ε(0)), (87)

we have to control, for a short time t < t0 ( in [0, t0) all the series in the game are absolutely

converging) the difference of the two series

N−1∑n=0

αεn(j)Tεn(j) (88)

∞∑n=0

Tεn(j). (89)

To estimate the error, we realize that (88) is truncated at N − 1. However the series is

absolutely geometrically convergent and hence the truncation error is bounded byλN

1− λ,

for λ < 1, so this error is very small. Another error arises from the difference between

αεn(j) and 1.

We have

1− αεn(j) ≤ 1−(

1−(N − j − n+ 1

N

)n)≤ n(j + n)

N(90)

26

Furthermore, since we have proved that for t sufficiently small |Tn(j)| ≤ Cjλn with λ < 1,

|N−1∑n=0

(1− αεn(j))Tn(j)| ≤N−1∑n=0

Cjλnn(j + n)

N≤ Cjε

2. (91)

Therefore it remains to estimate

N−1∑n=0

αεn(j)(Tεn(j)− Tεn(j)). (92)

Observe now that Tεn and Tεn differ for two reasons. The dynamics ζε(s)) and ζε(s)

agree only if recollisions are absent. Secondly the initial data f ε,0n+j and f0,n+j are different.

To simplify the notation, from now on we shall consider the case j = 1.

To control the recollisions we introduce the quantity, defined for a fixed tree G(1, n)

and a fixed sequence σ:

Rεn(z,G(1, n), σ, t) =

∫dΛ(tn, ωn,V1,n)Bε

nχrecf ε0,j+n(ζ

ε(0)) (93)

where χrec = 1 if and only if |ξεi (s)− ξεj (s)| < ε for some pair i 6= j and some time s ∈ [0, t).

Note that this includes the overlapping at time 0.

We now show that, under suitable hypotheses (see (95) below)

Rεn(z,G(1, n), σ, t) ≤ Cnn3εα

tn−1

(n− 1)!(94)

for any α < 1.

We assume (57) and

sup(Xj ,Vj)∈Γεj

|f ε0,j(Xj, Vj)− f ε0,j(Xj, Vj)| ≤ εCje−βV2j . (95)

Proof of (94)

We first observe that

χrec ≤n∑k=1

∑i<j

χki,j

where χki,j = χki,j(tn, ωn,V1,n) denotes the characteristic function of the event for which,

during the evolution of the particle system (given by the free dynamics ζε(s)!) particle i

and particle j overlap, for the first time, in the time interval (tk, tk+1), k = 0 . . . n, (t0 =

t, tn+1 = 0).

27

Then, thanks to (57)

Rεn(z,G(1, n), σ, t) ≤ Rε

n,1 +Rεn,2

where

Rεn,1 =

zn+1

n∑k=0

∑i<j

∫dΛ(tn, ωn,V1,n)Bε

nχki,jχ({|Wi,j(tk)| >

2√β| log ε|1/2}) exp

{−β

n+1∑r=1

|ηr(0)|2},

(96)

Rεn,2 =

zn+1

n∑k=0

∑i<j

∫dΛ(tn, ωn,V1,n)Bε

nχki,jχ({|Wi,j(tk)| ≤

2√β| log ε|1/2}) exp

{−β

n+1∑r=1

|ηr(0)|2}.

(97)

Here Wi,j(tk) = ηj(tk)− ηi(tk).We start by estimating Rε

n,1.

By the energy conservation

χ({|Wi,j(tk)| >2√β| log ε|1/2}) exp

{−β

2

n+1∑r=1

|ηr(0)|2}≤

χ({|ηj(tk)|2 + |ηi(tk)|2 >2

β| log ε|}) exp

{−β

2

n+1∑r=1

|ηr(tk)|2}≤ ε.

Note that we only use the square root of the Gaussian decay in velocities entering in

the definitions (96) and (97), since the other square root has to be used to control the

integrals in velocities. Therefore, by using the same arguments of the previous section, we

conclude that

Rεn,1 ≤ Cnn3ε

tn

n!. (98)

The estimate of Rεn,2 requires geometrical considerations.

Let th, h ≤ k be the the time of the last interaction of i or j (remind that i < j). More

precisely time th ≥ tj is the minimum time in which particle i or particle j are created or

create new particles. There are two possibilities.

Case 1. th is the birthday time of the particle j, namely th = tj.

Case 2. h > j hence at time th particle i or j creates particle h.

28

Obviously in the time interval (tk, th) the motions of i and j are free and the overlapping

between i and j, by construction, happens in (tk+1, tk).

Denote

W = ηj(t−h )− ηi(t−h )

and

Y = ξj(th)− ξi(t−h ).

The recollision condition implies

|PW⊥Y | =|W ∧ Y ||W |

≤ ε (99)

where PW⊥ = 1 − W⊗W|W |2 is the projection on the plane orthogonal to W . Indeed the time

s∗ minimizing |Y −Ws| is s∗ = Y ·W|W |2 .

Consider Case 1. Denoting by k the progenitor of j. Setting

W ′ = ηk(t+j )− ηi(t+j )

then

Y = Y0 +W ′tj

where

Y0 = ξi(tj−1)− ξk(tj−1)− σjεωj −W ′tj−1.

Therefore|W ∧W ′tj +W ∧ Y0|

|W |≤ ε.

Since Y0 is independent of tj so that we can integrate on tj with the above constraint. The

result of this integration produces

Cε1

|W ∧W ′|

where W = W|W | is the unit vector in the direction of W . To make the above expression

integrable with respect to the velocity variables (using that χki,j ≤ 1) we bound the time

integral more conveniently by

Cµεµ

(1

|W ∧W ′|

)µ

.

where µ < 1.

Consider now Case 2.

29

Assuming that the particle h has been created by j (the case in which it is created by

i is perfectly the same), we set

W ′ = ηj(t+h )− ηi(t+h ) = ηj(t

−h−1)− ηi(t−h−1)

then

Y = Y0 +W ′th

where

Y0 = ξi(th−1)− ξj(th−1)−W ′th−1.

The recollision condition is again

|W ∧W ′th +W ∧ Y0||W |

≤ ε.

The integration on th with the above constraint yields

Cµεµ(

1

|W ∧W ′|µ).

In conclusion

Rεn ≤ Cnn3εµ| log ε|1/2 tn−1

(n− 1)!(100)

Note that we have to replace n by n− 1 because we have used a time integration and

the presence of | log ε|1/2 is due to the bound on the relative velocity. �

To control the difference Tεn(j)− Tεn(j) we consider∫dΛ(tn, ωn,V1,n)

[Bεng

0n+1(ζε(0))−Bε

nfε,0n+1(ζε(0))

](101)

for any choice of G(1, n) ∈ G(1, n) and σn, is vanishing as ε → 0. To see this, we split

(101) into two terms:∫dΛ(tn, ωn,V1,n)

[Bεnf0,n+1(ζ

ε(0))−Bε

nf0,n+1(ζε(0))]

(102)

and ∫dΛ(tn, ωn,V1,n)Bε

n

[f0,n+1(ζε(0))− f ε,0n+1(ζε(0))

](103)

Since ζε(0)) 6= ζε(0)) only in case of recollisions, then, by (94), (102) is bounded by

Cnn3εαtn−1

(n− 1)!

30

while, thanks to (57), using the same arguments as for the Lanford’s Theorem, (103) is

bounded by

Cnεtn

n!.

Therefore we have established the following

Theorem 3 Suppose f0 ∈ C1. Under hypotheses (57), (58) and (56), there exists

t0 > 0 such that, for t ≤ t0 and for all j = 1, 2, . . . the following estimate holds true for

any α < 1 and a.a. Zj

|f εj (Zj, t)− fj(Zj, t)| ≤ C(j, α)εα. (104)

2.8 Law of large numbers

In the physical texbooks dealing with kinetic theory (and also in our introduction), the

quantity

f(x, v, t)dxdv (105)

is interpreted as the fraction of the total number of molecules falling in the cell of the phase

space around the point (x, v), of volume dxdv .

On the other hand, in deriving the Boltzmann equation, the quantity (105) has a

sligthly different meaning, namely it denotes the (a-priori) probability of finding a given

particle in that cell.

It is natural to investigate whether the two concepts agree in the present context in the

Boltzmann-Grad limit. The mathematical way to formulate this problem is to take the so

called empirical measure, defined by

µ(dz, t) =1

N

N∑i=1

δ(z − zi(t))dz. (106)

where

ZN(t) = Φt(ZN) = {z1(t) . . . zN(t)}

is the hard-sphere flow, and to see whether

µ(dz, t)→ f(z, t) weakly (107)

as ε→ 0, where f(z, t) solves the Boltzmann equation. By (107) we mean

1

N

N∑i=1

ϕ(zi(t))→ (ϕ, f(t)) (108)

31

for all continuous bounded ϕ. Here (f, g) =∫dxdvf(x, v)g(x, v).

We first remark that, due to the time irreversibility of the Boltzmann equation, we

cannot hope that the following claim is true:

If1

N

N∑i=1

ϕ(zi)→ (ϕ, f0) (109)

then (108) holds at time t > 0.

Actually this claim is false because it contradicts the H-theorem.

Suppose, by absurdum the claim is true. Then define, for any Borel measure ν in the

one-particle phase space, the operator R by∫(Rν(dxdv))ϕ(x.v) =

∫ν(dxdv)ϕ(x,−v)

ϕ ∈ Cb, namely R inverts the velocity.

Next denote by Bt the operator solving the Boltzmann equation and BtN the corre-

sponding operator for the N - particle system. Then∫BtNµNϕ(x.v) =

1

N

N∑i=1

ϕ(zi(t))→ (ϕ, f(t))

Clearly, we have

BtNRB

tNµN = µN .

On the other hand, if the claim were true, we have also

BtRBtf0 = f0. (110)

But the above identity cannot be true (if f0 is not a Maxwellian) because H(Rf) = H(f)

so that the entropy of the left hand side of (110) must be strictly smaller than the entropy

of the right hand side.

This means that we have to loose something in the description of the system before

and after the limit.

Let us come back to the problem od establishing a law of large number to give a sense

to the asymptotic equivalence between µN(t) and f(t). We have

Proposition 4

Under the hypotheses of the Lanford’s theorem, for any ϕ ∈ Cb and η > 0,

limε→0

PrN(|µN(ϕ, t)− (ϕ, f(t))| > η) = 0. (111)

32

Here PrN denotes the probability computed by means of WN0 and µN(ϕ, t) = 1

N

∑Ni=1 ϕ(zi(t)).

Proof.

By the Chebyshev inquality

PrN(|µN(ϕ, t)− (ϕ, f(t))| > η) ≤ 1

η2EN

(|µN(ϕ, t)− (ϕ, f(t))|2

)(112)

We evaluate the right hand side of (112) by using the Lanford’s theorem

EN(µN(ϕ, t)) = EN(1

N

N∑i=1

ϕ(zi(t))) =

∫f ε(z, t)ϕ(z)→ (ϕ, f(t)) (113)

and

EN(µN(ϕ, t)2) =1

N2

N∑i1=1

N∑i2=1i1 6=i2

EN(ϕ(zi1(t))ϕ(zi2(t))) +1

N

N∑i=1

EN(ϕ(zi(t))2) (114)

=1

N(N − 1)

∫f ε(z1, z2, t)ϕ(z1)ϕ(z2) +

1

N

∫f ε(z, t)ϕ(z)2

→ (ϕ, f(t))

Expanding the square in the right hand side of (112) and using (113) and (114) we

readly obtain (111). �

It would be interesting to formulate and prove a strong law of large numbers with a

quantitative explicit estimate. This is not yet known.

3. The Boltzmann eqution for soft potentials

3.1 Low-density limit

In this chapter we want to derive the Boltzmann equation for a particle system interact-

ing via a two-body, smooth, short-range potential. In this occasion we also present the

Boltzmann-Grad limit in a different perspective which may clarify its physical meaning.

Our starting point is a system of N identical point particles of unitary mass, interacting

by means of a two-body potential φ. We assume that the interaction potential φ is positive,

smooth and with range one: namely φ(x) ≥ 0 and φ(x) = 0 if |x| > 1 .

33

Denoting as before by (x1, . . . , xN ; v1, . . . , vN) a state of the system, where xi and vi

indicate the position and the velocity of the particle i, the dynamical flow is obtained by

solving the Newton equations

xi(t) =∑j 6=i

F (xi(t)− xj(t)) (115)

where F (xi − xj) = Fi,j = −∇φ(xi − xj) is the force due to the particle j, acting on the

particle i. We assume φ spherically symmetric.

As usual we consider a probability distribution WN of the system which is initially

(and hence at any positive time) symmetric in the exchange of the particles.

Its time evolution is described by the Liouville equation, which reads as:

(∂t + LN)fN = 0, (116)

where the Liouville operator LN is

LN = L0N + LI

N (117)

with:

L0N =

N∑i=1

vi · ∇xi (118)

and

LIN =

N∑i=1

∑i 6=j

Fi,j · ∇vi (119)

and Fi,j = −∇φ(xi − xj).We know that the marginals fNj (x1, v1, . . . xj, vj, t) of the time evolved measureWN(ZN , t) =

WN(XN , VN , t), satisfy the BBKGY hierarchy.

Consider now a small parameter ε indicating the ratio between macro and micro unities.

For instance the inverse of the number of Angstrom necessary to fill a meter (namey 10−10).

We pass to macroscopic variables defining

ξ = εx; τ = εt

When the macroscopic variables are O(1), the times which we are considering from a

microscopic point of view are very large. In this scale of times a tagged particles (say

particle 1) must deliver a finite number of collisions. As consequence the density Nε3 → 0

must vanish.

34

To be more precise we introduce (normalized) distribution functions of a j- particle

subsystem in term of macroscopic variables:

gNj (ξ1, v1, . . . ξj, vj) = ε−3j fNj (ε−1ξi, v1, . . . ε−1ξj, vj) (120)

where fNj are the distribution functions (the marginals of WN), expressed in microscopic

variables. In terms of the new variables, the BBKGY hierarchy becomes:

(∂τ +

j∑i=1

vi · ∇ξi)gNj +

1

ε

j∑i=1

∑k 6=i

F (ξi − ξkε

) · ∇vigNj (121)

= −N − jε

∫dξj+1

∫dvj+1F (

ξi − ξj+1

ε) · ∇vj+1

gNj+1.

For a fixed j the interaction term in the left hand side of eq.n (121) is negligible in the

L1(ξ1, v1 . . . ξj, vj) sense. Moreover the integral in the right hand side is O(ε3). Therefore

the interaction part of the group of the first j particles with the rest of the system is O(1)

whenever N = O(ε−2). This is the scaling which we are considering, namely

N →∞, ε→ 0, Nε2 = λ−1 > 0, (122)

for a system of N particles interacting via a smooth, short range potential of range ε.

Note that this is exactly the scaling we have postulated for the hard-core system and

it is usually called a low density limit or, in a macroscopic variables, the Boltzmann-Grad

limit.

3.2 The Grad’s hierarchy

We write down once more the Liouville equation (116) in macroscopic variables (denoted

however in the more familiar way as x, v, t) with the agreement that the range of the

potential is ε and that N and ε are related by the scaling law (122).

However, instead of considering the usual BBKGY hierarchy, we derive another more

convenient equation for a sequence of quantities slightly differing from the marginal distri-

butions, but asymptotically equivalent to these. This new hierarchy is more related to the

BBKGY hierarchy for hard-spheres and is more handable to the purpouse of deriving the

Boltzmann equation.

From now on we will reduce to a single parameter ε, being N = ε−2 as for the hard-

sphere case.

We define

f εj (Xj, Vj, t) =

∫S(Xj)N−j

dzj+1 . . . dzN WN(Xj, Vj, zj+1 . . . ZN , t). (123)

35

where (Xj, Vj) = (x1v1 . . . xj.vj) and

S(Xj) = {z = (x, v)||x− xk| > ε for all k = 1 . . . j}. (124)

It is clear that the functions fj, for any j, are asymptotically equivalent (in the L1 sense

in the limit ε→ 0) to the usual marginals

gεj (Xj, Vj, t) =

∫dzj+1 . . . dzN W

N(Xj, Vj, zj+1 . . . ZN , t). (125)

Consider now a configuration ZN = (Zj, Zj,N) where Zj = (z1 . . . zj) and Zj,N =

(zj+1 . . . zN) and where zi = (xi, vi). When

|x` − xk| > ε

for all ` = 1 . . . j and k = j+ 1 . . . N , since the range of the interaction is ε, the interaction

between the group of the first j particles and the rest of the system is vanishing. Therefore

the Liouville equation becomes:

∂tWN + LI

jWN + LI

N−jWN +

N∑i=1

vi · ∇xiWN = 0, (126)

where LIN−j is the interaction part of the Liouville operator relative to the last N − j

particles. Integrating eq.n (126) with respect to dzj+1 . . . dzN we obtain:

(∂tWNj + LI

j )WNj (x1, v1 . . . xj, vj) = −

∫S(Xj)N−j

dzj+1 . . . dzN (127)

N∑i=1

vi · ∇xiWN(x1, v1 . . . xj, vj, xj+1, vj+1 . . . xN , vN).

Here we used that: ∫dvj+1 . . . dvNLI

N−jWN = 0. (128)

Next we split the sum

N∑i=1

vi · ∇xiWN =

(N∑

i=j+1

+

j∑i=1

)vi · ∇xif

N (129)

in eq.n (127).

The first sum is handled by the divergence theorem yielding

−N∑

k=j+1

∫S(Xj)

dzj+1 . . .

∫∂S(Xj)

dσ(xk)dvk . . .

∫S(Xj)

dzN (130)

(vk · nk)WN(Xj, Vj, XN−j, VN−j).

36

where nk is the outward normal to S(Xj) and σ(dxk)dvk is the surface measure. Note that

∂S(Xj), as regards the x-dependence, is the union of spherical surfaces:

∂S(Xj) =

j⋃i=1

σi(Xj)× R3 (131)

where

σi(Xj) ⊂ {x| |x− xi| = ε}. (132)

Indeed if zk ∈ ∂S(Xj), there exists i ∈ {1, . . . , j}, an index of a particle of the first group,

such that |xi− xk| = ε. Note that there is only one of such an index for almost all xk with

respect to the surface measure σ(dxk). We also set ni,k = xi−xk|xi−xk|

. Using the symmetry of

WN we have N − j identical integrals for which (130) becomes:

−(N − j)j∑i=1

∫σi(Xj)

dσ(xj+1)

∫dvj+1(vj+1 · ni,j+1) (133)∫

S(Xj)N−j−1

WN(Xj, Vj, xj+1, vj+1, zj+2 . . . zN)dzj+2 . . . dzN .

Unfortunately the integration domain of last integral in (133) is not S(Xj+1)N−j−1, as

it would be necessary to recover fj+1 and close the equation. Note however that reducing

this integration to S(Xj+1)N−j−1 would produce a small error. Nevertheless we want to

establish an exact equation for which we need a further work.

Let I be a subset of indices of {1, . . . N}. We denote by |I| its cardinality. In particulare

we denote J = {1, . . . j} and Jk,N = {k+ 1, . . . N} (for which |J | = j and |Jk,N | = N − k).

Then S(Xj)N−j−1 can be expressed in terms of the following union of disjoint sets

S(Xj)N−j−1 =

N−j−1⋃k=1

⋃I⊂Jj+1,N :|I|=k

∆I(Xj+1)× S(x1 . . . xj+1, XI)N−j−1−k. (134)

Here XI = {xi1 . . . xik} where I = {i1 . . . ik} and

∆I(Xj+1) ⊂ S(Xj)k

is a set for which there exists an ordering

I = {i1 . . . ik}

such that

|xi1 − xj+1| ≤ ε, |xir − xr+1| ≤ ε, r = 1 . . . k − 1. (135)

37

This decomposition means the following. A generic configuration in S(Xj)N−j−1 differs

from S(Xj+1)N−j−1 because some particle could overlap with particle j + 1. If it is the

case, we consider the maximal chain of overlapping particles (those with indices I) . The

other particles are far apart the group with indices J ∪ {j + 1} ∪ I, therefore each of them

is in S(x1 . . . xj+1, XI). As a consequence we can decompose the integration domain as the

union of all such events to obtain

(133) =− (N − j)j∑i=1

N−j−1∑k=0

∑I⊂Jj+1,N :|I|=k

∫σi(Xj)

dσ(xj+1)

∫dvj+1(vj+1 · ni,j+1) (136)

∫∆I(Xj+1)

dZI

∫S(x1...xj+1,XI)N−j−1−k

dZLWN(Xj, Vj, xj+1, dvj+1, ZI , ZL).

Here L denotes the set of indices {j + 2 . . . N}/I.

We can simplify the above expression by using the symmetry of WN obtaining

(133) =−j∑i=1

N−j−1∑k=0

(N − j)(N − j − 1) . . . (N − j − k) (137)∫σi(Xj)

dσ(xj+1)

∫dvj+1(vj+1 · ni,j+1)∫

∆j,k

dzj+2 . . . dzNfεj+k+1(Xj, Vj, xj+1vj+1, zj+2 . . . zj+1+k),

where

∆j,k = {zj+2 . . . zj+k+1| |xi − xi−1| < ε, i = j + 2, . . . j + k + 1}.

To treat the second sum we consider

(Vj · ∇Xj)WNj (Xj, Vj) = (Vj · ∇Xj)

∫SN−j(Xj)

dXN−jdVN−jWN(Xj, Vj, XN−j, VN−j) (138)

=d

dt

∫SN−j(Xj+Vjt)

dXN−jdVN−jWN(Xj + Vjt, Vj, XN−j, VN−j)

∣∣t=0

=

∫SN−j(Xj)

dXN−jdVN−j(Vj · ∇Xj)WN(Xj, Vj, XN−j, VN−j)+

− (N − j)j∑i=1

∫σi(Xj)

dσ(xj+1)

∫dvj+1(vi · ni,j+1)∫

S(Xj)N−j−1

WN(Xj, Vj, xj+1, vj+1, zj+2 . . . zN)dzj+2 . . . dzN .

38

Repeating the same step we did before to obtain (137) we readly arrive to the following

hierarchy of equations:

∂tfNj + Lε

jfNj =

N−j−1∑k=0

Aj+k+1fεj+k+1, j ≤ N (139)

where

Aj+k+1fεj+k =

j∑i=1

(N − j)(N − j − 1) . . . (N − j − k)ε2 (140)∫dn

∫dvj+1(vi − vj+1) · n∫

∆j,k

dzj+2 . . . dzNfεj+k+1(Xj, Vj, xj+1vj+1, zj+2 . . . zj+1+k).

Note that

Aj+1fεj+1(Xj, Vj) =

j∑i=1

(N − j)ε2

∫dn

∫dvj+1(vi − vj+1) · nf εj+1(Xj, Vj, xi + nε, vj+1)

(141)

= (N − j)ε2Cεj+1f

εj+1(Xj, Vj),

where Cεj+1 is the same collision operator appearing in the hard-sphere analysis.

Actually this term is the only O(1) term in the sum in the right hand side of (139).

Indeed, for k > 0 and fixed j, the size of the Aj+k+1 is

O(Nk+1ε2ε3k)

the ε3k term coming from the successive integrations in the domain ∆j,k. Thus we are in

a situation quite similar to that of the hard spheres case and, therefore, we can hope to

derive the Boltzmann equation in a similar manner.

Actually this can be done and we can derive for the one-particle distribution the fol-

lowing Boltzmann equation

(∂t + v · ∇)f = Q(f, f), (142)

where

Q(f, f)(x, v) =

∫dv1

∫S+

(v − v1) · n[f(x, v′)f(x, v′1)− f(x, v)f(x, v1)]. (143)

39

Note that this is not the same as the Boltzmann equation for hard spheres in spite of

its apparent similarity. To see this, let us consider the two-body problem, associated to

the potenyial φ, which is almost explicitly solvable. The two-body problem can be reduced

to a central collision passing to the variables:

V = v1 − v2, W =v1 + v2

2(144)

that are the relative velocity and the velocity of the center of mass. We assume to work

in an inertial frame for which W = 0. The relative velocity satisfies the Newton equation

(with the reduced mass 12) associated to the same potential φ. If V is the velocity before

the collision, the outgoing velocity V ′ (that is the velocity after the collision, when the

particle exits from the sphere of radius ε in which the potential differs from zero) is given

by:

V ′ = V − 2ω(V · ω) (145)

and ω is the angle which bisects the two directions of V and V ′. This follows by the energy

and angular momentum conservation. Remember that the motion takes place in the plane

identified by the origin and the vector V . If α is the angle formed by the vectors V and ω,

then θ = π − 2α is called the scattering angle.

Hence V ′ is determined by ω (or by n or the scattering angle θ). Note that the vector

n is the versor of the line joining the origin to the point in the sphere of radius ε where the

particle is starting to interact, namely the entrance point in the interaction disk. ω is the

versor of the line joining the origin with the point of the minimal distance from the origin.

Obviously n and ω are not the same unless for the hard-sphere interaction. It is more

convenient to express n · V in terms of ω because V ′ (and hence v′1 and v′2 ) are naturally

expressed in term of ω.

Passing to the two-body problem we can therefore determine the outgoing relative

velocity and, by the conservation laws, we finally arrive to the expression for the outgoing

velocities v′1, v′2:

v′1 = v1 − ω(v1 − v2 · ω) (146)

v′2 = v2 + ω(v1 − v2 · ω).

To express V · n in terms of ω, we observe now that |V ·n||V | dn is the fraction of particles

scattered in the solid angle corresponding to dω(n), which is, by definition, the cross section

σ(|V |, ω)dω. Setting B(|V |, ω) = |V |σ(|V |, ω), we can write the collision operator of the

Boltzmann equation associated to our potential as

Q(f, f)(x, v1) =

∫dv2

∫S+

dωB(|V |, ω)[f(x, v′1)f(x, v′2)− f(x, v1)f(x, v2)]. (147)

40

It is possible to bound the collision operator and perform a tree expansion as we did

for the hard-sphere system, however we do not give here the details.

Remark

From the beginning of kinetic theory it arised the problem of computing B, the main

problem being the behavior of B with the relative velocity V = v1 − v2. A more tractable

case is that of potential obeyng a power law

φ(x) =1

|x|s

for an integer s > 0. In this case the behavior of B in terms of |V | is explicitely known.

However the above potential is long-range. The versor n does not make sense as well the

gain and loss terms, which are diverging if taken separately.

There are recent results concerning the Boltzmann equation with these cross-sections

(for which the gain and loss term compensate because the divergence happens on the

grazing collisions where where they are practically the same). However it is very doubtful

if the Boltzmann equation is valid for long-range potentials.

ER

41

4. The Landau equation

4.1 Grazing collision limit

The Boltzmann equation works for a rarefied gas, however one can ask whether a useful

kinetic picture can be invoked also for a dense gas. Using the same notation as the previous

chapter, we consider a particle system (directly in macroscopic variables) interacting via a

two-body smooth interaction φ with range ε.

We set N = O(ε−3) to express that the gas is dense, but the particles are weakly

interacting. To express the weakness of the interaction, we assume that the two-body

potential φ is rescaled by√ε. Since φ varies on a scale ε (in macro unities), the force

is O( 1√ε) but acts on the time interval O(ε). The momentum variation due to the single

scattering is therefore O(√ε). The number of particles met by a test particles is O(1

ε).

Hence the total momentum variation for unit time is O( 1√ε). However this variation, in

case of homogeneous gas and symmetric force, should be zero in the average. If we compute

the variance, it should be 1εO(√ε)2 = O(1). As a consequence of this central limit type of

argument we expect that the kinetic equation which holds in the limit (if any), should be

a diffusion equation in the velocity variable.

A more convincing argument will be presented in the next section. For the moment let

us now argue at level of kinetic equation. Consider the collision operator in the form

Q(f, f) =

∫dv1

∫dp w(p)δ(p2 + (v − v1) · p)[f ′f ′1 − ff1] (148)

where

f ′ = f(v + p), f ′1 = f(v1 − p)

being p the transferred momentum in the collision. Here the smooth kernel w, which mod-

ulates the collision, is assumed depending only on p trough its modulus (it is spherically

symmetric). The δ appearing in eq.n (148) is a consequence of the energy conservation.

Suppose that ε > 0 is a small parameter. To express the fact that the transferred mo-

mentum is small, we rescale w as 1ε3w(p

ε). In addition we also rescale the mean-free path

inverse by a factor 1ε

to take into account the high density situation. The collision operator

42

becomes:

Qε(f, f) =1

ε4

∫dv1

∫dp w(

p

ε)δ(p2 + (v − v1) · p)[f ′f ′1 − ff1] = (149)

1

2πε2

∫dv1

∫dp w(p)

∫ +∞

−∞dseis(p

2ε+(v−v1)·p)

[f(v + εp)f(v1 − εp)− f(v)f(v1)] =

1

2πε

∫dv1

∫dp w(p)

∫ 1

0

dλ

∫ +∞

−∞dseis(p

2ε+(v−v1)·p)

d

dλf(v + ελp)f(v1 − ελp) =

1

2πε

∫dv1

∫dp w(p)

∫ 1

0

dλ

∫ +∞

−∞dseis(p

2ε+(v−v1)·p)

p · (∇v −∇v1)f(v + ελp)f(v1 − ελp).

To outline the behavior of Qε(f, f) in the limit ε → 0, we introduce a test function ϕ

for which, after a change of variables ( here (·, ·) denotes the scalar product in L2(v)):

(ϕ,Qε(f, f)) =1

2πε

∫dv

∫dv1

∫dp w(p)

∫ 1

0

dλ

∫ +∞

−∞dseis(p

2(ε−2ελ)+(v−v1)·p) (150)

ϕ(v − ελp) p · (∇v −∇v1)ff1 =

1

2πε

∫dv

∫dv1

∫dp w(p)

∫ 1

0

dλ

∫ +∞

−∞dseis(v−v1)·p

[ϕ(v)− εp · ∇vϕ(v)] p · (∇v −∇v1)ff1+

1

2π

∫dv

∫dv1

∫dp w(p)

∫ +∞

−∞dseis(v−v1)·pϕ(v)

isp2

∫ 1

0

dλ(1− 2λ) p · (∇v −∇v1)ff1 +O(ε)

Note now that the term O(ε−1) vanishes because of the symmetry p→ −p (the function

p→ w(p)δ((v− v1) · p) is even)). Also the immaginary part of the O(1) term is vanishing,

being null the integral in dλ. As a result:

(ϕ,Qε(f, f)) = − 1

2π

∫dv

∫dv1

∫dp w(p)

∫ +∞

−∞ds eis(v−v1)·p (151)

p · ∇vϕ p · (∇v −∇v1)ff1 +O(ε).

Therefore we have recovered the kinetic equation

(∂t + v · ∇x)f = QL(f, f) (152)

43

with the new collision operator

QL(f, f) =

∫dv1∇v a(∇v −∇v1)ff1, (153)

where a = a(v − v1) denotes the matrix

ai,j(V ) =

∫dp w(p) δ(V · p) pipj. (154)

This matrix can be handled in a better way by introducing ther polar coordinates:

ai,j(V ) =1

|V |

∫dp |p|w(p) δ(V · p) pipj (155)

=B

|V |

∫dp δ(V · p) pipj,

where V and P are the versor of V and p respectively and

B =

∫ +∞

0

drr3w(r). (156)

Note that B is the only parameter describing the interaction appearing in the equation.

Finally a straightforward computation yields:

ai,j(V ) =B

|V |(δi,j − ViVj). (157)

The collision operator QL has been introduced by Landau in 1936 for the study of a

weakly intercting dense plasma.

From the mathematical side very little is known about the landau equation (152), even

for the homogeneous case. The main difficulty is due to the presence of the diverging factor1|V | in (153).

The qualitative properties of the solutions to the Landau equation are the same as for

the Boltzmann equation as regards the basic conservation laws and the H-theorem. Indeed

it is easy to show that

(vα, QL(f, f)) = 0 (158)

for α = 0, 1, 2 and the Entropy production is given by the following expression

−(log f,QL(f, f)) =1

2

∫dv

∫dv1

1

ff1

1

|v − v1||P⊥v−v1(∇v −∇v1)ff1|2. (159)

However, in the present lecture, we are mostly interested in deriving the Landau equa-

tion in terms of particle systems. In the next section we present a formal derivation

outlining the difficulties in trying a rigorous proof even for short times.

44

4.2 Weak-coupling limit for classical systems

We consider a classical system of N identical particles of unitary mass. Positions and

velocities are denoted by q1 . . . qN and v1 . . . vN . The Newton equations reads as:

d

dτqi = vi

d

dτvi =

∑j=1...N :j 6=i

F (qi − qj). (160)

Here F = −∇φ denotes the interparticle (conservative) force, φ the smooth, two-body,

spherically symmetric interaction potential and τ the time.

We are interested in a situation where the number of particles N is very large and the

interaction strength quite moderate. In addition we look for a reduced or macroscopic

description of the system. Namely if q and τ refer to the system seen in a microscopic

scale, we introduce ε > 0 a small parameter expressing the ratio between the macro and

microscales. Indeed it is often convenient to rescale eq.n (160) in terms of the macroscopic

variables

x = εq t = ετ

whenever the physical variables of interest are varying on such scales and are almost con-

stant on the microscopic scales. Therefore, rescaling the potential according to:

φ→√εφ, (161)

system (160), in terms of the (x, t) variables, becomes:

d

dtxi = vi

d

dtvi =

1√ε

∑j=1...N :j 6=i

F (xi − xjε

). (162)

Note that the velocities are authomatically unscaled. Moreover we also assume that

N = O(ε−3), namely the density is O(1).

Let WN = WN(XN , VN) be a symmetric (in the exchange of particles) probability

distribution on the phase space of the system. Here (XN , VN) denote the set of position

and velocities:

XN = x1 . . . xN VN = v1 . . . vN .

Then from eq.ns (162) we obtain the following Liouville equation

(∂t + VN · ∇N)WN(XN , VN) =1√ε

(T εNW

N)(XN , VN) (163)

45

where VN · ∇N =∑N

i=1 vi · ∇xi and (∂t + VN · ∇N) is the usual free stream operator and

(T εNWN)(XN , VN) =

∑0<k<`≤N

(T εk,`WN)(XN , VN), (164)

with

T εk,`WN = ∇φ(

xk − x`ε

) · (∇vk −∇v`)WN . (165)

From the Liouville equation (163) we have the usual BBKGY hierarchy of equations

for the marginals fNj (for 1 ≤ j ≤ N):

(∂t +

j∑k=1

vk · ∇k)fNj =

1√εT εj f

Nj +

N − j√εCεj+1f

Nj+1. (166)

The operator Cεj+1 is defined as:

Cεj+1 =

j∑k=1

Cεk,j+1 , (167)

and

Cεk,j+1fj+1(x1 . . . xj; v1 . . . vj) = (168)

−∫dxj+1

∫dvj+1F (

xk − x`ε

) · ∇vkfj+1(x1, x2, . . . , xj+1; v1, . . . , vj+1).

We finally fix the initial value {f 0j }Nj=1 of the solution {fNj (t)}Nj=1 assuming that {f 0

j }Nj=1

is factorized, that is, for all j = 1, . . . N

f 0j = f⊗j0 , (169)

where f0 is a given one-particle distribution function. Of course the statistical independence

at time t = 0 is destroyed at time t > 0 because dynamics creates correlations and eq.n

(166) shows that the time evolution of fN1 is determined by the knowledge of fN2 which turns

out to be dependent on fN3 and so on. However, since the interaction between two given

particle is going to vanish in the limit ε→ 0, we can hope that such statistical independence

is recovered in the same limit. Note that the physical meaning of the propagation of chaos

here is quite different from that arising in the context of the Boltzmann equation. Here two

particles can interct but the effect of the collision is small, while in a low-density regime

the effect of a collision between two given particles is large but quite unlikely.

Therefore we expect that in the limit ε → 0 the one-particle distribution function fN1converges to the solution of a suitable nonlinear kinetic equation f which we are going to

investigate.

46

By using the Duhamel formula, perurbing around the free flow S(t) defined as

(S(t)fj)(Xj, Vj) = fj(Xj − Vjt, Vj), (170)

we find

fNj (t) =S(t)f 0j +

N − j√ε

∫ t

0

S(t− t1)Cεj+1f

Nj+1(t1)dt1+ (171)

1√ε

∫ t

0

S(t− t1)T εj fNj (t1)dt1.

We now try to keep informations on the limit behavior of fNj (t). Assuming for the moment

that the time evolved j-particle distributions fNj (t) are smooth (in the sense that the

derivatives are uniformly bounded in ε), then

Cεj+1f

Nj+1(Xj;Vj; t1) = (172)

− ε3

j∑k=1

∫dr

∫dvj+1F (r) · ∇vkfj+1(Xj, xk − εr;Vj, vj+1, t1).

Assuming now, quite reasonably, that∫drF (r) = 0, (173)

we find that

Cεj+1f

Nj+1(Xj;Vj; t1) = O(ε4)

provided that D2vf

Nj+1 is uniformly bounded. Since

N − j√ε

= O(ε72 )

we see that the second term in the right hand side of (171) does not give any contribution

in the limit.

Moreover ∫ t

0

S(t− t1)T εj fNj (t1)dt1 = (174)∑

i 6=k

∫ t

0

dt1F((xi − xk)− (vi − vk)(t− t1)

ε

)g(Xj, Vj; t1)

where g is a smooth function.

Obviously the above time integral is O(ε) so that also the last term in the right hand

side of (171) does not give any contribution in the limit. Then we are facing the alternative:

47

either the limit is trivial or the time evolved distributions are not smooth. This is indeed

a bed new because, if we believe that the limit is not trivial (actually we expect to get the

Landau equation, according to the previous discussion) a rigorous proof of this fact seems

to be hard.

In order to look for a (non-trivial) kinetic equation, we can conjecture that

fNj = gNj + γNj (175)

where gNj is the main part of fNj and is smooth, while γNj is small, but strongly oscillating.

We operate this decomposition according to the following equations which define gNj and

γNj :

(∂t +

j∑k=1

vk · ∇xk)gNj =

N − j√εCεj+1g

Nj+1 +

N − j√εCεj+1γ

Nj+1 (176)

(∂t +

j∑k=1

vk · ∇xk)γNj =

1√εT εj γ

Nj +

1√εT εj g

Nj , (177)

with initial data

gNj (Xj, Vj) = f 0j (Xj, Vj), γNj (Xj, Vj) = 0. (178)

Note that γN1 = 0 since T ε1 = 0.

The remarkable fact of this decomposition is that γ can be eliminated. Indeed, let

(Xj(t), Vj(t)) = ({x1(t) . . . xj(t), v1(t) . . . vj(t)}) be the solution of the j-particle flow (in

macro variables)d

dtxi = vi

d

dtvi = − 1√

ε

∑k=1...j:k 6=i

∇φ(xi − xk

ε), (179)

with initial datum (Xj, Vj) = ({x1 . . . xj, v1 . . . vj}). Denote by Uj(t) (we skip the ε depen-

dence for notational simplicity) the operator solving the Liouville equation

(∂t + Vj · ∇j)h(Xj, Vj; t) =1√ε

(T εj h

)(XN , VN ; t) (180)

associated to the flow (179), namely

h(Xj, Vj, t) = Ujh(Xj, Vj) = h(Xj(−t), Vj(−t)). (181)

Then eq.n (177) can be solved:

γNj (t) = − 1√ε

∫ t

0

dsU(s)T εj gNj (t− s). (182)

48

or, more explicitely

γNj (Xj, Vj, t) = − 1√ε

∫ t

0

ds∑

1≤i<k≤j

∇φ(xi(−s)− xk(−s)

ε)·(∇vi−∇vk)g

Nj (Xj(−s), Vj(−s); t−s).

(183)

Inserting (183) in (176) we finally arrive to a closed hierarchy for gN which is

(∂t +

j∑k=1

vk · ∇xk)gNj (Xj, Vj; t) =

N − j√εCεj+1g

Nj+1(Xj, Vj; t)+

N − jε

j∑k=1

j+1∑i,r=1

∫ t

0

ds

∫dvj+1

∫dxj+1

divvkF (xk − xj+1

ε)F (

xi(−s)− xr(−s)ε

)(∇vi −∇vr)gNj+1(Xj+1(−s), Vj+1(−s); t− s). (184)

Now (Xj+1(t), Vj+1(t)) refers to the flow (179) for j + 1. Here and in the following the

combination F (x)F (y) = F (x)⊗ F (y) stands for the matrix FαFβ, α, β = 1, 2, 3.

The price we pay for getting a closed equation is to have a long tail memory effect in

the right hand side of (184).

A rigorous analysis of the limit N → ∞, ε = N−(1/3) seems to be very difficult. We

expect that, in this limit, fNj (t) and gNj (t) would converge to f(t)⊗j, where f solves the

Landau equation.

Remark Why we expect that γNj strongly oscillates?

Consider eq.n (180) with a smooth initial datum h0 and let us try to control the first

derivatives of h(Xj, Vj, t) = Uj(t)h(Xj, Vj) = h0(Xj(−t), Vj(−t)). We have

∂h(t)

∂xαi=∑k,β

(∂h0

∂xβk

∂xβk(−t)∂xαi

+∂h0

∂vβk

∂vβk (−t)∂xαi

)

and analogous formula for ∂h(t)∂vαi

. Here we are using Greek indices for the components of xi

and vi. To estimate quantities like∂xβk (−t)∂xαi

,∂xβk (−t)∂vαi

,∂vβk (−t)∂xαi

,∂vβk (−t)∂vαi

we use eq.n (179) and find,

by standard arguments, that we cannot get, in general, a behavior better than O(exp( 1√ε)).

On the other hand γNj is also expected to be small, in some sense. Indeed by taking the

scalar product of (182) by a smooth function u, we find

|(u, γNj (t))| ≤ 1√ε

∫ t

0

ds‖Uj(−s)u‖L∞‖T εj gNj (t− s)‖L1

≤ ε5/2 j(j − 1)

2‖u‖L∞

∫ t

0

ds

∫dx1

∫dx3 . . .

∫dVj

∫dr|F (r)|

|(∇v1 −∇v2)gNj (x1, x1 + εr, x3 . . . , Vj; t− s)|.

49

Therefore this term is vanishing provided that gN is sufficientlt smooth (uniformly in ε.

Of course we are not able to show any kind of smoothness on the solution of eq.n (184).

ER

Now we analyze the first equation of (184), assuming gN2 smooth. We have

(∂t + v1 · ∇x1)gN1 (t) =

N − 1√εCε

2gN2 (t) (185)

+N − 1

εCε

2

∫ t

0

dsU2(s)T2gN2 (t− s).

Let u ∈ D be a test function. By the same argument we have seen before, we conclude

thatN − 1

ε(u,Cε

2gN2 (t)) = O(

√ε).

Next we analyze the last term in the right hand side of (185). It is

N − 1

ε

∫dx1

∫dx2

∫dv1

∫dv2

∫ t

0

ds divu(x1, v1)

F (x1 − x2

ε)F (

xi(−s)− xk(−s)ε

) · (∇v1 −∇v2)gN2 (x1, x2, v1, v2; t− s) ≈ (186)∫

dx1

∫dr

∫dv1

∫dv2

∫ ∞0

ds divu(x1, v1)

F (r)F (xi(−εs)− xk(−εs)

ε) · (∇v1 −∇v2)g

N2 (x1, x2, v1, v2; t).

(r = x1−x2

ε) and s → s

ε. Observe now that, setting w = v1 − v2 the relative velocity, we

have

(x1(−εs)− x2(−εs)ε

) = r + ws+1

ε

∫ −εs0

dτ(v1(τ)− v1)− (v2(τ)− v2). (187)

On the other hand, we have

v1(τ)− v1 =1√ε

∫ τ

0

dsF ((x1(s)− x2(s)

ε) = O(

√ε), (188)

because the time spent when the two particles are at distance less that ε is O(ε), (if the

relative velocity w not too small). Thus, in conclusion

r.h.s.(185) ≈∫dx1

∫dr

∫dv1

∫dv2

∫ ∞0

ds divu(x1, v1)F (r)F (r + ws) (189)

(∇v1 −∇v2)gN2 (x1, x1, v1, v2; t)

≈(u,QL(gN1 , gN1 ).

50

In the last step we just were assuming the propagation of chaos, namely

gN2 ≈ gN,⊗21 .

Of course all the assumptions we made have to be proven in order to have a rigorous

proof of the Landau equation. This is, at moment, a difficult task.

Comments

A few comments on the bibliography. For generalities concerning the Boltzmann equa-

tion, see the popular monograph [9] and, for a more mathematical oriented text [10].

[8] is a remarkable biography of L. Boltzmann, including the collocation of is work in

the scientific community of his time.

The validity of the Boltzmann equation for hard spheres and for short times is due

to O. Lanford [20] (see also [10] for additional comments. The same analysis for pos-

itive potentials is due to F. King [18]. The starting point is the Grad’s papers where

the Boltzmann-Grad limit has been established as well the first equation of the BBKGY

hierarchy for soft potentials (which we call Grad’s hierarchy) (see [15] and [16]).

There is a case in which the Boltzmann equation can be validated for arbitrary times

[19], that is when the initial datum is a small perturbation of the vacuum. There are not

other results valid for large times.

The most general existence theorem for the initial value problem associated to the Boltz-

mann equation is the famous paper by Di Perna and Lions [12]. Other more constructive

results can be found in [10] and references quoted therein.

The Landau equation was originally derived by Landau in a grazing collision limit (see

[21]). For a mathematical treatment of this problem, see [1], [14], [24], [11] and [25].

An heuristic direct connection with particle system in the weak-coupling limit is pre-

sented in [2]. See also [22].

The compatibility of the Landau equation in the weak-coupling limit with the particle

system (a sort of derivation at time zero) has been proved in [3].

The situation for quantum systems (not discussed in the present notes) can be con-

sidered very preliminary. The problem can be formulated for the Wigner transform [26],

which describe quantum states in the classical phase space. Here the weak-coupling limit,

as well the low density limit, give a Boltzmann equation (with different cross-sections in

the two cases). For the weak-coupling limit, if one consider the statistics, the form of the

Boltzmann equation is more complicated [23] because it includes trilinear terms.

From the mathematical side there are very partial results, [17], [4], [5], [6] , [7] and [13].

The problem of a complete derivation is still open, even for short times.

51

References

[1] A.A. Arseniev, O.E. Buryak, On a connection between the solution of the Boltzmann

equation and the solution of the Landau-Fokker-Planck equation. (Russian) Mat. Sb

181 (1990), no. 4, 435–446; translation in Math. USSR-Sb. 69 (1991), no. 2, 465–478

[2] R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics, John Wiley & Sons,

New-York (1975).

[3] A. Bobylev, M. Pulvirenti, C. Saffirio (in preparation)

[4] D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti, Some Considerations on the

derivation of the nonlinear Quantum Boltzmann Equation, J. Stat. Phys. 116, Nos 114

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