The Boltzmann-Grad limit of the periodic Lorentz gas

0.71 MB·English

by  Jens Marklof

THE BOLTZMANN-GRAD LIMIT OF THE PERIODIC LORENTZ GAS
JENS MARKLOFANDANDREASSTRO¨MBERGSSON
Abstract. We study the dynamics of a point particle in a periodic array of spherical scat-
terers, and construct a stochastic process that governs the time evolution for random initial
data in the limit of low scatterer density (Boltzmann-Grad limit). A generic path of the
limiting process is a piecewise linear curve whose consecutive segments are generated by a
Markov process with memory two.
8
0
0
2

n
a
J Contents

3
1. Introduction 2

] 1.1. The joint distribution of path segments 2
S
1.2. A limiting stochastic process for the billiard flow 4
D
1.3. Macroscopic initial conditions 5
.
h 1.4. Outline of the paper 7
t
a 2. First collision 7
m 2.1. Location of the first collision 7
[ 2.2. Scattering maps 9

2.3. Velocity after the first impact 10
1
v 2.4. A uniform version of Theorem 2.2 11
2 3. Iterated scattering maps 17
1
3.1. Two lemmas 17
6
0 3.2. The main proposition 20
. 4. Loss of memory 23
1
0 4.1. Statement of the main theorem 23
8 4.2. Sets of good initial velocities 24
0
4.3. Proof of Theorem 4.1 27
:
v 4.4. Proof of Theorem 1.1 33
i
X 4.5. Proof of Theorem 1.3 34
r 5. Convergence to the stochastic process Ξ(t) 36
a
5.1. Four lemmas 36
5.2. Proof of Theorem 5.1 39
5.3. A counterexample 40
5.4. Macroscopic initial conditions 40
6. A continuous-time Markov process 41
6.1. Transcription of Theorem 1.4 41
6.2. A semigroup of propagators 43
6.3. The Fokker-Planck-Kolmogorov equation 45
References 50
Date: 3 January 2008.
J.M. has been supported by EPSRC Research Grants GR/T28058/01 and GR/S87461/01, and a Philip
Leverhulme Prize. A.S. is a Royal Swedish Academy of Sciences Research Fellow supported by a grant from
theKnutand Alice Wallenberg Foundation.
1
2 JENSMARKLOFANDANDREASSTRO¨MBERGSSON
1. Introduction
TheLorentz gas describes an ensemble of non-interacting pointparticles in an infinitearray
ofsphericalscatterers. ItwasoriginallydevelopedbyLorentz[18]in1905tomodel,inthelimit
of low scatterer density (Boltzmann-Grad limit), the stochastic properties of the motion of
electrons inametal. Inthepresentpaperweconsiderthecaseofaperiodicarrayofscatterers,
and construct a stochastic process that indeed governs the macroscopic dynamics of a particle
cloud in the Boltzmann-Grad limit. The corresponding result has been known for some time
inthecaseofaPoisson-distributed(rather thanperiodic)configuration ofscatterers. Herethe
limitingprocesscorrespondstoasolutionofthelinearBoltzmannequation,seeGallavotti [13],
Spohn [26], and Boldrighini, Bunimovich and Sinai [4]. It already follows from the estimates
in [5, 16] that the linear Boltzmann equation does not hold in the periodic set-up; this was
pointed out recently by Golse [14, 15].
Our results complement classical studies in ergodic theory that characterize the stochastic
properties of the periodic Lorentz gas in the limit of long times, see [6, 2, 9, 21, 22, 1, 25, 12]
for details.
To state our main results, consider an ensemble of non-interacting point particles moving in
an array of spherical scatterers which are placed at the vertices of a euclidean lattice Rd
L ⊂
of covolume one (Figure 1). The dynamics of each particle is governed by the billiard flow
(1.1) ϕ :T1( ) T1( ), (q ,v ) (q(t),v(t))
t Kρ → Kρ 0 0 7→
where Rd is the complement of the set d + (the “billiard domain”), and T1( ) =
Kρ ⊂ Bρ L Kρ
Sd 1 is its unit tangent bundle (the “phase space”). d denotes the open ball of radius
Kρ× 1− Bρ
ρ, centered at the origin. A point in T1( ) is parametrized by (q,v), with q denoting
ρ ρ
K ∈ K
the position and v Sd 1 the velocity of the particle. The Liouville measure on T1( ) is
∈ 1− Kρ
(1.2) dν(q,v) = dvolRd(q)dvolS1d−1(v)
where volRd and volSd1−1 refer to the Lebesgue measures on Rd and Sd1−1, respectively. For the
purpose of this introduction we will restrict our attention to Lorentz’ classical set-up, where
the scatterers are assumed to be hard spheres. Our results in fact also hold for scattering
processes described by smooth potentials, see Sec. 2.2 for details.
Iftheinitialcondition(q ,v )israndom,thebilliardflowgivesrisetothestochasticprocess
0 0
(1.3) (q(t),v(t)) :t R
>0
{ ∈ }
which we will refer to as the Lorentz process. The central result of this paper is the existence
of a limiting stochastic process Ξ(t) :t R of theLorentz process in the Boltzmann-Grad
>0
{ ∈ }
limit ρ 0. We begin with a study of the distribution of path segments of the billiard flow
→
between collisions.
1.1. The joint distribution of path segments. Thebilliard flow ϕ induces a billiard map
t
on the boundary ∂T1( ),
ρ
K
(1.4) (q ,v ) (q ,v ),
n−1 n−1 7→ n n
where q ,v denote position and velocity at the nth collision in the outgoing configuration,
n n
i.e.,
(1.5) (q ,v ) = lim ϕ (q ,v ).
n n ǫ→0+ τ1(qn−1,vn−1;ρ)+ǫ n−1 n−1
Here τ denotes the free path length, defined by
1
(1.6) τ (q,v;ρ) = inf t > 0 : q+tv / .
1 ρ
{ ∈ K }
We will later also use the parametrization ∂T1( ) = (Sd 1+ ) Sd 1, so that q = m +
Kρ ρ− L × 1− n n
ρw , where w Sd 1 and m are the position on the ball and ball label at the nth
n n ∈ 1− n ∈ L
collision.
THE BOLTZMANN-GRAD LIMIT OF THE PERIODIC LORENTZ GAS 3
q v Q V
0 0 0 0
s S
1 1
s S
2 2
2ρ fixed 2ρd const ρd 1
−
×
Figure 1. Left: The periodic Lorentz gas in “microscopic” coordinates—the
lattice remainsfixedastheradiusρofthescatterertendstozero. Right: The
L
periodic Lorentz gas in “macroscopic” coordinates —both the lattice constant
and the radius of each scatter tend to zero, in such a way that the mean free
path length remains finite. The vectors s ,s ,... (resp. S ,S ,...) represent
1 2 1 2
the segments of the billiard path between collisions.
The time elapsed between the (n 1)th and nth hit is defined as the nth collision time
−
(1.7) τ (q ,v ;ρ) = τ (q ,v ;ρ).
n 0 0 1 n 1 n 1
− −
We express the nth path segment by the vector
(1.8) s (q ,v ;ρ) := τ (q ,v ;ρ)v (q ,v ;ρ).
n 0 0 n 0 0 n 1 0 0
−
The central result of [19] is the proof of a limiting distribution for the first collision time
τ , and further refined versions that also take into account the particle’s direction after the
1
reflection. We will here extend these results to find a joint limiting distribution for the first n
segments of the billiard path with initial coordinates (q ,v ), where the position q is fixed
0 0 0
and the velocity v random. The precise statement is the following.
0
Here and in the remainder of this paper we will use the standard representation = ZdM ,
0
L
where M SL(d,R). We will also use the notation S := S 1S. We set
0 −
∈ k k
(1.9) := (S ,...,S ) (Rd 0 )n : S = S (j =1,...,n 1) .
n 1 n j+1 j
B ∈ \{ } b 6 −
Theorem 1.1. Fix(cid:8)a lattice = ZdM and a point q Rd , and write α(cid:9) = q M 1.
L 0 b 0 ∈b \L − 0 0−
Then for each n Z there exists a function P(n) : R such that, for any Borel
>0 α n 0
probability measur∈e λ on Sd 1 which is absolutely continBuou→s wit≥h respect to vol , and for
1− S1d−1
any set Rnd with boundary of Lebesgue measure zero,
A⊂
(1.10) limλ v Sd 1 :(s (q ,v ;ρ),...,s (q ,v ;ρ)) ρ (d 1)
ρ 0 0 ∈ 1− 1 0 0 n 0 0 ∈ − − A
→
(cid:0)(cid:8) (cid:9)(cid:1)
(n)
= Pα (S1,...,Sn)λ′(S1)dvolRd(S1) dvolRd(Sn),
···
ZA
where λ′ ∈ L1(Sd1−1) is the Radon-Nikodym derivative of λ wbith respect to volS1d−1. Further-
more, there is a function Ψ : R such that
3 0
B → ≥
n
(n) (2)
(1.11) P (S ,...,S ) = P (S ,S ) Ψ(S ,S ,S )
α 1 n α 1 2 j 2 j 1 j
− −
j=3
Y
for all n 3 and all (S ,...,S ) .
1 n n
≥ ∈ B
4 JENSMARKLOFANDANDREASSTRO¨MBERGSSON
Theaboveconditionq Rd ensuresthatτ isdefinedforρsufficientlysmall. InSec.4.4
0 1
∈ \L
we also consider variants of Theorem 1.1 where the initial position is near , e.g., q ∂ .
L 0 ∈ Kρ
We define the probability measure corresponding to (1.10) by
(n) (n)
(1.12) µα,λ(A):= Pα (S1,...,Sn)λ′(S1)dvolRd(S1)···dvolRd(Sn).
Z
A
Note in particular that µ(n+1)( Rd) = µ(n)( )b.
α,λ A× α,λ A
Remark 1.1. In probabilistic terms, Theorem 1.1 states that the discrete-time stochastic pro-
cess ρd 1s (q ,v ;ρ) :n Z converges in the limit ρ 0 to
{ − n 0 0 ∈ >0} →
(1.13) S : n Z ,
n >0
{ ∈ }
a Markov process with memory two. As we shall see, Ψ(S ,S ,S ) is in fact independent of
1 2 3
S .
1
k k
(n) (n)
Remark 1.2. If d 3 then P is continuous on all of . If d = 2 then P is continuous
α n α
≥ B
except possibly at points (S ,...,S ) with S = S or S = S R for some
1 n ∈ Bn 2 − 1 j+2 j Sj+1
1 j n 2, where R O(2) denotes reflection in the line RS. Cf. Remark 4.7 below.
S
≤ ≤ − ∈
b b b b
(n)
Remark 1.3. Note that Ψ is independent of and q , and P depends only on the choice of
L 0 α
(n)
α. This means in particular that Ψ and P are rotation-invariant, i.e., for any K O(d) we
α
∈
have
(1.14) Ψ(S K,S K,S K)= Ψ(S ,S ,S )
1 2 3 1 2 3
and
(n) (n)
(1.15) P (S K,...,S K) =P (S ,...,S ).
α 1 n α 1 n
For α Rd Qd also P(n) =:P(n) is in fact independent of α, cf. Remark 4.6 below. Explicit
α
∈ \
formulas and asymptotic properties of the limiting distributions will be presented in [20].
Remark 1.4. Thecasen= 1ofcourseleadstothedistributionofthefreepathlengthdiscussed
in [19]; cf. also [10, 5, 16, 7, 3] for earlier results.
1.2. A limiting stochastic process for the billiard flow. In Theorem 1.1 we have iden-
tified a Markov process with memory two that describes the limiting distribution of billiard
paths with random initial data (q ,v ). Let us denote by
0 0
(1.16) Ξ(t) : t R ,
>0
{ ∈ }
the continuous-time stochastic process that is obtained by moving with unit speed along the
random paths of the Markov process (1.13). The process is fully specified by the probability
(1.17) P Ξ(t ) ,...,Ξ(t )
α,λ 1 1 M M
∈D ∈ D
that Ξ(t) visits the sets ,..., (cid:0) T1(Rd) at times t = t(cid:1),...,t , with M arbitrarily
1 M 1 M
large. To give a precise DdefinitioDn of⊂(1.17) set T := 0, T := n S , and define the
0 n j=1k jk
probability that Ξ(t) is in the set at time t after exactly n hits, in the set at time t
1 1 1 2 2
D P D
after exactly n hits, etc., by
2
(1.18) P(n) Ξ(t ) ,...,Ξ(t ) and T t < T ,...,T t < T
α,λ 1 ∈ D1 M ∈ DM n1 ≤ 1 n1+1 nM ≤ M nM+1
(n+1)
(cid:0) := µ (S ,...,S ) :Ξ (t ) , T t < T (j = 1,...,M)(cid:1)
α,λ 1 n+1 nj j ∈Dj nj ≤ j nj+1
with n := (n ,...,n ),(cid:0)(cid:8)n := max(n ,...,n ), and (cid:9)(cid:1)
1 M 1 M
n
(1.19) Ξ (t) := S +(t T )S ,S .
n j n n+1 n+1
−
(cid:18)j=1 (cid:19)
X
b b
THE BOLTZMANN-GRAD LIMIT OF THE PERIODIC LORENTZ GAS 5
Note that the choice T t < T of semi-open intervals is determined by the use of the
n n+1
≤
outgoing configuration, recall (1.5). The formal definition of (1.17) is thus
(1.20) P Ξ(t ) ,...,Ξ(t )
α,λ 1 1 M M
∈ D ∈ D
:= P(n(cid:0)) Ξ(t ) ,...,Ξ(t ) (cid:1) and T t < T ,...,T t < T .
α,λ 1 ∈ D1 M ∈ DM n1 ≤ 1 n1+1 nM ≤ M nM+1
nX∈ZM≥0 (cid:0) (cid:1)
The following theorem shows that the Lorentz process (1.3), suitably rescaled, converges
to the stochastic process (1.16) as ρ 0. Given any set T1(Rd) we say that t 0 is
→ D ⊂ ≥
-admissible if
D
(1.21) vol S Sd 1 : (tS ,S ) ∂ = 0.
Sd1−1 1 ∈ 1− 1 1 ∈ D
We write adm( ) for the set of(cid:0)a(cid:8)ll -admissible numbers t (cid:9)0(cid:1).
b b b
D D ≥
Theorem 1.2. Fix a lattice = ZdM and a point q Rd , set α= q M 1, and let λ
L 0 0 ∈ \L − 0 0−
be a Borel probability measure on Sd 1 which is absolutely continuous with respect to vol .
1− S1d−1
Then, for any subsets ,..., T1(Rd) with boundary of Lebesgue measure zero, and any
1 M
D D ⊂
numbers t adm( ) (j = 1,...,M),
j j
∈ D
(1.22) limλ v Sd 1 :(ρd 1q(ρ (d 1)t ),v(ρ (d 1)t )) , j = 1,...,M
ρ 0 0 ∈ 1− − − − j − − j ∈ Dj
→
(cid:0)(cid:8) = P Ξ(t ) ,...,Ξ(t(cid:9)(cid:1)) .
α,λ 1 1 M M
∈ D ∈ D
The convergence is uniform for (t ,...,t ) in compact subse(cid:0)ts of adm( ) adm( (cid:1) ).
1 M 1 M
D ×···× D
Remark 1.5. The condition t adm( ) cannot be disposed with. For example, (1.22) is in
j j
∈ D
general false in the case M = 1, = d Sd 1. We prove this in Section 5.3. Note however
D1 Bt1× 1−
that no admissibility condition is required in the macroscopic analogue of Theorem 1.2, see
Theorem 1.4 below.
1.3. Macroscopic initialconditions. Inviewoftherescalingappliedintheprevioussection
it is natural to consider the “macroscopic” billiard flow
(1.23) F :T1(ρd 1 ) T1(ρd 1 )
t − ρ − ρ
K → K
(Q ,V ) (Q(t),V(t)) = (ρd 1q(ρ (d 1)t),v(ρ (d 1)t)),
0 0 7→ − − − − −
and take random initial conditions (Q ,V ) with respect to some fixed probability measure
0 0
Λ. We will establish the analogous limit laws as in the previous sections. Although the
macroscopic versions are less general (they are obtained by averaging over q ), they appear
0
more natural from a physical viewpoint, where one is interested in the time evolution of a
macroscopic particle cloud; cf. the discussion at end of this section.
The nth path segment in these macroscopic coordinates is
(1.24) S (Q ,V ;ρ) := ρd 1s (ρ (d 1)Q ,V ;ρ).
n 0 0 − n − − 0 0
Theorem 1.3. Fix a lattice and let Λ be a Borel probability measure on T1(Rd) which is
L
absolutely continuous with respect to Lebesgue measure. Then, for each n Z , and for any
>0
∈
set Rd Rnd with boundary of Lebesgue measure zero,
A ⊂ ×
(1.25) limΛ (Q ,V ) T1(ρd 1 ): (Q ,S (Q ,V ;ρ),...,S (Q ,V ;ρ))
ρ 0 0 0 ∈ − Kρ 0 1 0 0 n 0 0 ∈ A
→
(cid:0)(cid:8) (cid:9)(cid:1)
= P(n)(S1,...,Sn)Λ′ Q0,S1 dvolRd(Q0)dvolRd(S1)···dvolRd(Sn),
ZA
(cid:0) (cid:1)
with P(n) as in Remark 1.3, and where Λ is the Rbadon-Nikodym derivative of Λ with respect
′
to volRd×volSd1−1.
6 JENSMARKLOFANDANDREASSTRO¨MBERGSSON
The probability measure corresponding to the above limiting distribution is defined by
(1.26) µΛ(n)(A):= P(n)(S1,...,Sn)Λ′ Q0,S1 dvolRd(Q0)dvolRd(S1)···dvolRd(Sn).
ZA
We redefine the stochastic process (1.16)(cid:0)by spec(cid:1)ifying the probability
b
(1.27) P Ξ(t ) ,...,Ξ(t )
Λ 1 1 M M
∈ D ∈ D
(n)
via the measure µ by the same(cid:0)construction as in Section 1.2(cid:1). The only essential difference
Λ
is that we need to replace (1.19) by
n
(1.28) Ξ (t) := Q + S +(t T )S ,S .
n 0 j − n n+1 n+1
(cid:18) j=1 (cid:19)
X
Note that formally P = P if Λ(Q,V) = δ(Q)λ(V),band αb Rd Qd.
Λ α,λ ′ ′
∈ \
Theorem 1.4. Fix a lattice and let Λ be a Borel probability measure on T1(Rd) which is
L
absolutely continuous with respect to Lebesgue measure. Then, for any t ,...,t R , and
1 M 0
any subsets ,..., T1(Rd) with boundary of Lebesgue measure zero, ∈ ≥
1 M
D D ⊂
(1.29) limΛ (Q ,V ) T1(ρd 1 ): (Q(t ),V(t )) ,...,(Q(t ),V(t ))
ρ 0 0 0 ∈ − Kρ 1 1 ∈ D1 M M ∈DM
→
(cid:0)(cid:8) = P Ξ(t ) ,...,Ξ(t ) (cid:9)(cid:1).
Λ 1 1 M M
∈ D ∈ D
The convergence is uniform for t ,...,t in compact subsets of R .
1 M (cid:0) 0 (cid:1)
≥
The time evolution of an initial particle cloud f L1(T1(ρd 1 )) in the periodic Lorentz
− ρ
∈ K
gas is described by the operator L defined by
t,ρ
(1.30) [L f](Q,V)= f(F 1(Q,V)).
t,ρ t−
To allow a ρ-independent choice of the initial density f, it is convenient to extend the action
of F from T1(ρd 1 ) to T1(Rd) by setting F = id on T1(Rd) T1(ρd 1 ). We fix the
t − ρ t − ρ
K \ K
Liouville measure on T1(Rd) to be the standard Lebesgue measure
(1.31) dν(Q,V) = dvolRd(Q)dvolS1d−1(V).
Theorem1.4now impliestheexistence of alimiting operator L thatdescribestheevolution of
t
theparticle cloud intheBoltzmann-Grad limit. Moreprecisely, forevery set withboundary
D
of Lebesgue measure zero, we have
(1.32) lim [L f](Q,V)dν(Q,V)= [L f](Q,V)dν(Q,V),
t,ρ t
ρ 0
→ ZD ZD
uniformly for t on compacta in R , and L is defined by the relation
0 t
≥
(1.33) [L f](Q,V)dν(Q,V)= P Ξ(t) ,
t Λ
∈ D
ZD
for any absolutely continuous Λ, any Borel subset T(cid:0)1(Rd), an(cid:1)d f = Λ. We note that L
′ t
D ⊂
commutes with the translation operators T :R Rd ,
R
{ ∈ }
(1.34) [T f](Q,V) := f(Q R,V),
R
−
and, in view of Remark 1.3, with the rotation operators R :K O(d) ,
K
{ ∈ }
(1.35) [R f](Q,V):= f(QK,VK).
K
ItwasalreadypointedoutbyGolse[15],thattheweak- limitofanyconvergingsubsequence
∗
L f (ρ 0) does not satisfy the linear Boltzmann equation. His arguments usethe a priori
t,ρi i →
estimates in [5, 16], and do not require knowledge of the existence of the limit (1.32). The
fundamental reason behind the failure of the linear Boltzmann equation is that, perhaps
surprisingly, L :t 0 is not a semigroup. We will show in Section 6 how to overcome this
t
{ ≥ }
problem by considering an extended stochastic process that keeps track not only of position
Q and current velocity V, but also the free path length until the next collision, and
T
THE BOLTZMANN-GRAD LIMIT OF THE PERIODIC LORENTZ GAS 7
the velocity V thereafter. We will establish that the extended process is Markovian, and
+
derive the corresponding Fokker-Planck-Kolmogorov equation describing the evolution of the
particle density in the extended phase space. A similar approach has recently been explored
by Caglioti and Golse in the two-dimensional case [8]. Their result is however conditional on
an independence hypothesis, which is equivalent to the Markov property established by our
Theorem 1.3 above.
1.4. Outline of the paper. The key ingredient in the present work is Theorem 4.8 of [19]
(restated as Theorem 2.2 below for general scattering maps), which yields the joint limiting
distribution for the free path length and velocity after the next collision, given that the initial
position and velocity are taken at random with respect to a fixed probability measure. The
proofsof Theorems 1.1 and1.3 arebasedon auniformversion of Theorem2.2, wherethefixed
probability measures are replaced by certain equismooth families, see Section 2 for details.
Section 3 provides technical information on the nth iterate of the scattering maps, which in
conjunction with the uniform version of Theorem 2.2 yields the proof of Theorems 1.1 and 1.3
(Section 4). In Section 5 we prove that the dynamics in the periodic Lorentz gas converges
in the Boltzmann-Grad limit to a stochastic process Ξ(t), and thus establish Theorems 1.2
and 1.4. We finally derive the substitute for the linear Boltzmann equation in Section 6, by
extending Ξ(t) to a Markov process and calculating its Fokker-Planck-Kolmogorov equation.
2. First collision
We begin by reviewing the central result of [19].
2.1. Location of the first collision. We fix a lattice = ZdM with M SL(d,R), once
0 0
L ∈
and for all. Recall that Rd is the complement of the set d+ and that the free path
Kρ ⊂ Bρ L
length for the initial condition (q,v) T1( ) is defined as
ρ
∈ K
(2.1) τ (q,v;ρ) = inf t > 0 : q+tv / .
1 ρ
{ ∈ K }
Note that τ (q,v;ρ) = can only happen for a set of v’s of measure zero with respect to
1
∞
vol . In factwe have τ (q,v;ρ) < wheneverthedcoordinates of vM 1 Rd arelinearly
Sd1−1 1 ∞ 0− ∈
independentover Q (note that this condition is independentof q), since then each orbit of the
linear flow x x+tv is dense on Rd/ .
7→ L
The position of the particle when hitting the first scatterer is
(2.2) q (q,v;ρ) := q+τ (q,v;ρ)v.
1 1
As in Sec. 1.1 we write q (q,v;ρ) = m +ρw with m and w = w (q,v;ρ) Sd 1.
1 1 1 1 ∈ L 1 1 ∈ 1−
Let us fix a map K : Sd 1 SO(d) such that vK(v) = e for all v Sd 1; we assume
1− → 1 ∈ 1−
that K is smooth when restricted to Sd 1 minus one point. For example, we may choose K
1−
as K(e ) = I, K( e )= I and
1 1
− −
2arcsin v e /2
1
(2.3) K(v) = E k − k v
− v ⊥
(cid:16) (cid:0)k ⊥k (cid:1) (cid:17)
for v Sd 1 e , e , where
∈ 1− \{ 1 − 1}
0 w
(2.4) v := (v ,...,v ) Rd 1, E(w)= exp SO(d).
⊥ 2 d ∈ − tw 0 ∈
(cid:18)− (cid:19)
Then K is smooth when restricted to Sd 1 e .
1− \{− 1}
It is evident that −w1K(v) ∈ S′1d−1, with the hemisphere
(2.5) S′1d−1 = {v = (v1,...,vd) ∈ Sd1−1 : v1 > 0}.
Letβ beacontinuous functionSd 1 Rd. Ifq weassumethat(β(v)+R v) d =
1− → ∈ L >0 ∩B1 ∅
for all v Sd 1. We will consider initial conditions of the form (q (v),v) T1( ), where
∈ 1− ρ,β ∈ Kρ
8 JENSMARKLOFANDANDREASSTRO¨MBERGSSON
q (v) = q +ρβ(v) and where v is picked at random in Sd 1. Note that for fixed q and β
ρ,β 1−
we indeed have q (v) for all v Sd 1, so long as ρ is sufficiently small.
ρ,β ∈ Kρ ∈ 1−
For the statement of the theorem below, we recall the definition of the manifolds X (y)
q
and X(y) from [19, Sec. 7]: If q Z and α q 1Zd, then we set X = Γ(q) SL(d,R), and
>0 − q
∈ ∈ \
define, for each y Rd 0 ,
∈ \{ }
(2.6) X (y) := M X : y (Zd+α)M .
q q
∈ ∈
We also set X = ASL(d,Z) ASL(d,(cid:8)R) where ASL(d,R) = SL((cid:9)d,R) ⋉ Rd is the semidirect
\
product group with multiplication law (M,ξ)(M ,ξ ) = (MM ,ξM +ξ ); we let ASL(d,R)
′ ′ ′ ′ ′
act on Rd through y y(M,ξ):= yM +ξ. Now for each y Rd we define
7→ ∈
(2.7) X(y) := g X : y Zdg .
∈ ∈
The spaces Xq(y) and X(y) carry natura(cid:8)l probability meas(cid:9)ures νy whose properties are dis-
cussed in [19, Sec. 7].
We will also use the notation
(2.8) x = x (x e )e , for x Rd.
1 1
⊥ − · ∈
The following is a restatement of [19, Theorem 4.4].
Theorem 2.1. Fix a lattice = ZdM . Let q Rd and α = qM 1. There exists a
L 0 ∈ − 0−
function Φ : R ( 0 d 1) ( 0 Rd 1) R such that for any Borel probability
measure λαon Sd1>−01×ab{so}lu×telBy1c−ont×inu{ou}s×with−resp→ect t≥o0volS1d−1, any subset U ⊂ S′1d−1 with
vol (∂U) = 0, and 0 ξ < ξ , we have
Sd1−1 ≤ 1 2
(2.9) limλ v Sd 1 : ρd 1τ (q (v),v;ρ) [ξ ,ξ ), w (q (v),v;ρ)K(v) U
ρ 0 ∈ 1− − 1 ρ,β ∈ 1 2 − 1 ρ,β ∈
→
(cid:0)(cid:8) ξ2 (cid:9)(cid:1)
= Φ ξ,w,(β(v)K(v)) dλ(v)dwdξ,
α
Zξ1 ZU ZS1d−1 ⊥
⊥ (cid:0) (cid:1)
where dw denotes the (d 1)-dimensional Lebesgue volume measure on 0 Rd 1. The
−
− { } ×
function Φ is explicitly given by
α
(2.10)
ν M X (y) : (Zd+α)M (Z(0,ξ,1)+z) = if α q 1Zd
y q −
Φ (ξ,w,z) = ∈ ∩ ∅ ∈
α (νy(cid:0)(cid:8)g ∈ X(y) : Zdg∩(Z(0,ξ,1)+z)= ∅ (cid:9)(cid:1) if α∈/ Qd,
where y = ξe +w+z(cid:0)(cid:8), and (cid:9)(cid:1)
1
(2.11) Z(c ,c ,σ) = (x ,...,x ) Rd : c < x < c , (x ,...,x ) < σ .
1 2 1 d 1 1 2 2 d
∈ k k
Remark 2.1. For α Qd(cid:8)the function Φ (ξ,w,z) is Borel measurable, an(cid:9)d in fact only
α
∈
depends on α and the four real numbers ξ, w , z ,z w. Also for α Qd, if we restrict to
k k k k · ∈
z 1 [and if d = 2: z+w = 0], then Φ (ξ,w,z) is jointly continuous in the three variables
α
k k ≤ 6
ξ,w,z. If α / Qd then Φ (ξ,w,z) is everywhere continuous in the three variables, and it is
α
∈
independent of both α and z; in fact it only depends on ξ and w . We have, for all α Rd
k k ∈
and all z 0 Rd 1,
−
∈ { }×
∞
(2.12) Φ (ξ,w,z)dwdξ = 1.
α
Z0 Z{0}×B1d−1
The convergence in this integral is uniform, i.e.
∞
(2.13) Φ (ξ,w,z)dwdξ 0
α
ZT Z{0}×B1d−1 →
uniformly with respect to α and z as T . Cf. [19, Remark 4.5, (8.37) and Lemma 8.15].
→ ∞
THE BOLTZMANN-GRAD LIMIT OF THE PERIODIC LORENTZ GAS 9
Remark 2.2. We may extend the function Φ (ξ,w,z) to the larger set R ( 0 d 1)
( 0 Rd 1) by letting Φ (0,w,z) := 1 for aαll w,z. This definition is na≥tu0r×al,{sin}c×e iBt1m−ake×s
− α
{ }×
Φ (ξ,w,z) continuous (jointly in the three variables) at each point with ξ = 0. The proof of
α
this is an immediate extension of the discussion in [19, Sec. 8.1, 8.2].
2.2. Scattering maps. As indicated above, the results of this paper extend to the case of
a Lorentz gas, where the scattering process at a hard sphere is replaced by a smooth radial
potential. ToobtainthecorrectscalingintheBoltzmann-Gradlimit, weassumethescattering
potential is of the form V(q/ρ), where V(q) has compact support in the unit ball d. We will
B1
refer to the rescaled ball d =ρ d as the interaction region.
Bρ B1
It is most convenient to phrase the required assumptions in terms of a scattering map that
describes the dynamics at each scatterer. Let
(2.14) := (v,w) Sd 1 Sd 1 : v w < 0
S− { ∈ 1− × 1− · }
be the set of incoming data (w ,v ), i.e., the relative position and velocity with which the
− −
particle enters the interaction region. Thecorrespondingoutgoing data is parametrized by the
set
(2.15) := (v,w) Sd 1 Sd 1 : v w > 0 .
S+ { ∈ 1− × 1− · }
We define the scattering map by
(2.16) Θ : , (v ,w ) (v ,w ).
+ + +
S− → S − − 7→
In the case of the original Lorentz gas the scattering map is given by specular reflection,
(2.17) Θ(v,w) = (v 2(v w)w,w);
− ·
scattering maps corresponding to smooth potentials can be readily obtained from classical
results, see e.g. [23, Chapter 5].
In the following we will treat the scattering process as instantaneous. In the case of po-
tentials the particle will of course spend a non-zero amount of time in the interaction region,
but—under standard assumptions on the potential—this time will tend to zero when ρ 0.
→
Let Θ (v,w) Sd 1 and Θ (v,w) Sd 1 be the projection of Θ(v,w) Sd 1 onto the
1 ∈ 1− 2 ∈ 1− ∈ 1−
first and second component, respectively. We assume throughout this paper that
(i) the scattering map Θ is spherically symmetric, i.e., if (v ,w ) = Θ(v,w) then
+ +
(v K,w K)= Θ(vK,wK) for all K O(d);
+ +
∈
(ii) v and w are contained in the subspace spanned by v and w;
+ +
(iii) if w = v then v = v;
+
− −
(iv) Θ : is C1 and for each fixed v Sd 1 the map w Θ (v,w) is a C1
diffeoSm−or→phSis+m from w Sd 1 : v w < 0∈on1t−o some open sub7→set o1f Sd 1.
{ ∈ 1− · } 1−
The above conditions are for example satisfied for the scattering map of a “muffin-tin”
Coulomb potential, V(q) = αmax( q 1 1,0) with α / 2E,0 , where E denotes the
−
k k − ∈ {− }
total energy. Conditions (iii) and (iv) help to simplify the presentation of the proofs, but are
not essential. It is for instance not necessary that in (iv) the map w Θ (v,w) is invertible
1
7→
as long as it has finitely many pre-images, which allows a larger class of scattering potentials;
condition (iii) can be dropped entirely.
We will write ϕ(v,u) [0,π] for the angle between any two vectors v,u Rd 0 . Using
∈ ∈ \{ }
thesphericalsymmetry and Θ (v, v) = v one sees that thereexists a constant 0 B < π
1 Θ
− − ≤
such that for each v Sd 1, the image of the diffeomorphism w Θ (v,w) is
∈ 1− 7→ 1
(2.18) := u Sd 1 : ϕ(v,u) > B .
Vv { ∈ 1− Θ}
Let us write β : w Sd 1 : v w < 0 for the inverse map. Then β is spherically
−v Vv → { ∈ 1− · } −v
symmetric in the sense that β (uK) = β (u)K for all v Sd 1, u , K O(d), and in
−vK −v ∈ 1− ∈Vv ∈
particular β (u) is jointly C1 in v,u.
−v
10 JENSMARKLOFANDANDREASSTRO¨MBERGSSON
We also define
(2.19) β+(u)= Θ (v,β (u)) (v Sd 1, u ).
v 2 −v ∈ 1− ∈ Vv
The map β+ is also spherically symmetric and jointly C1 in v,u. In terms of the original
scattering situation, thepointof ournotation is thefollowing: Given anyv ,v Sd 1, there
exists w ,w Sd 1 such that Θ(v ,w ) = (v ,w ) if and only if ϕ(v− ,v+ ∈) >1−B , and
− + ∈ 1− − − + + − + Θ
in this case w and w are uniquely determined, as w = β (v ).
+ ±v +
For example−, in the case of specular reflection (2.17)±we hav−e B = 0 and
Θ
v v
+
(2.20) w+ = w = − − .
− v+ v
k − −k
Remark 2.3. In the case of specular reflection, the flow F preserves the Liouville measure ν,
t
but this does not hold in the case of a general scattering map satisfying (i)–(iv). Indeed, a
necessary and sufficient condition for F to preserve ν is that the scattering map Θ is a diffeo-
t
morphism from onto which carries the volume measure v w dvol (v)dvol (w)
S− S+ | · | S1d−1 S1d−1
on to (v w)dvol (v)dvol (w) on . Maps with this property can be classified
S− · Sd1−1 Sd1−1 S+
explicitly: Define the functions ϑ ,ϑ : ( π, π) R through
1 2 −2 2 →
(2.21) Θ e , (cosϕ)e +(sinϕ)e = (cosϑ (ϕ))e +(sinϑ (ϕ))e .
j 1 1 2 j 1 j 2
− −
In view of (iii) we(cid:0)may then assume ϑ (0) =(cid:1)0, and take ϑ , ϑ to be continuous. (Then ϑ ,ϑ
j 1 2 1 2
are both odd and C1, and ϑ is a C1 diffeomorphism from ( π,π) onto (B π,π B ).)
1 −2 2 Θ − − Θ
In this notation, one checks by a computation that Θ carries v w dvol (v)dvol (w)
| · | S1d−1 S1d−1
to (v w)dvol (v)dvol (w) if and only if, for all ϕ ( π,π) 0 ,
· Sd1−1 Sd1−1 ∈ −2 2 \{ }
sin(ϑ (ϕ) ϑ (ϕ)) d 2 cos(ϑ (ϕ) ϑ (ϕ))
(2.22) 2 sin−ϕ 1 − · 2 cos−ϕ 1 · ϑ′2(ϕ)−ϑ′1(ϕ) = 1.
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12)
This is seen (cid:12)to hold if and only if(cid:12) ϑ (ϕ(cid:12)) = ϑ (ϕ) ϕ for all(cid:12)ϕ(cid:12) ( π, π) or ϑ(cid:12) (ϕ) = ϑ (ϕ)+ϕ
(cid:12) (cid:12) 2 (cid:12) 1 − (cid:12) ∈ −2 2 2 1
forallϕ ( π, π). (Inphysicalterms,thisreflectsthepreservationoftheangularmomentum
∈ −2 2
w v, or its reversal, respectively.) Translating this condition in terms of β , we conclude
±
∧
that F preserves the Liouville measure if and only if
t
(2.23) β+ (v ) β ( v ) or β+ (v ) β (v )R ,
v1 2 ≡ −−v2 − 1 v1 2 ≡ −v2 1 {v2}⊥
where R O(d) denotes orthogonal reflection in the hyperplane v Rd.
{v2}⊥ ∈ { 2}⊥ ⊂
2.3. Velocity after the first impact.
Theorem 2.2. Let λ be a Borel probability measure on Sd 1 absolutely continuous with respect
1−
to vol . For any bounded continuous function f :Sd 1 R Sd 1 R,
Sd1−1 1− × >0× 1− →
(2.24) lim f v ,ρd 1τ (q (v ),v ;ρ),v (q (v ),v ;ρ) dλ(v )
0 − 1 ρ,β 0 0 1 ρ,β 0 0 0
ρ→0ZSd1−1
(cid:0) (cid:1)
= f v ,ξ,v p (v ,ξ,v )dλ(v )dξdvol (v ),
ZSd1−1ZR>0ZS1d−1 0 1 α,β 0 1 0 S1d−1 1
(cid:0) (cid:1)
where the probability density p is defined by
α,β
Φ ξ,w,(β(v )K(v )) dw if v
(2.25) pα,β(v0,ξ,v1)dvolSd1−1(v1)= (0α(cid:0) 0 0 ⊥(cid:1) if v11 ∈∈/ VVvv00,
with
(2.26) w = β v K(v ) 0 d 1.
− −e1 1 0 ∈ { }×B1−
⊥
(cid:0) (cid:1)