Table Of ContentON THE DISTRIBUTION OF FREE PATH LENGTHS
FOR THE PERIODIC LORENTZ GAS III
3
0
0
EMANUELE CAGLIOTI AND FRANC¸OIS GOLSE
2
n
a Abstract. For r (0,1), let Zr = x R2 dist(x,Z2) > r/2
∈ { ∈ | }
J and τ (x,v) = inf t > 0 x+tv ∂Z . Let Φ (t) be the proba-
r { | ∈ r} r
5 bility thatτ (x,v) t forx andv uniformly distributedinZ and
r r
2 S1 respectively. We≥prove in this paper that
] 1 1/4 t dr 2 1
S limsup Φ = +O
D ǫ→0+ |lnǫ|Zǫ r(cid:18)r(cid:19) r π2t (cid:18)t2(cid:19)
. 1 1/4 t dr 2 1
ath liǫm→0in+f |lnǫ|Zǫ Φr(cid:18)r(cid:19) r = π2t +O(cid:18)t2(cid:19)
as t + . This result improves upon the bounds on Φ in
m r
→ ∞
Bourgain-Golse-Wennberg [Commun. Math. Phys. 190 (1998),
[
491–508]. We also discuss the applications of this result in the
1 context of kinetic theory.
v
0
0
3
1 1. Statement of the problem and main results
0
3 1.1. The periodic Lorentz gas. Let r (0, 1) and define
0 ∈ 2
/ (1.1) Z = x R2 dist(x,Z2) > r/2 .
h r
{ ∈ | }
t
a Consider a point particle moving at speed 1 inside Z and being specu-
r
m
larlyreflected each timeit meets theboundaryofZ . Such adynamical
r
:
v system is referred to as “a periodic, two-dimensional Lorentz gas”. (In-
i deed, Lorentzusedthemethodsofkinetictheorytodescribethemotion
X
of electrons in a metal as that of a collisionless gas of point particles
r
a bouncing on the crystalline structure of atoms in the metal [13]).
The “free path length” (or “(forward) exit time”) starting from x
∈
Z in the direction v S1” is defined as
r
∈
(1.2) τ (x,v) = inf t > 0 x+tv ∂Z , (x,v) Z S1.
r r r
{ | ∈ } ∈ ×
For each v = (v ,v ) S1 such that v v = 0 and the ratio v /v
1 2 1 2 1 2
∈ 6 ∈
R Q, one has τ (x,v) < + , since any orbit of a linear flow with
r
\ ∞
irrationalslopeonthe2-torusisdense (seeforinstance in[1], —section
51, corollary 1, p. 287 — for this well-known fact).
Set Y = Z /Z2; since τ (x,v) = τ (x+k,v) for each (x,v) Z S1
r r r r r
∈ ×
and k Z2, the function τ can be seen as defined on Y S1.
r r
∈ ×
1
2 E. CAGLIOTI ANDF. GOLSE
1 r
Figure 1. The Lorentz gas: Z and the punctured torus Y
r r
1.2. Invariant measure for the Lorentz gas. Let
1
V = dxdv meas(Y S1) and µ = dxdv.
r r r
− × V
r
Thus µ is a Borelian probability measure on Y S1.
r r
×
On the other hand, the evolution of the Lorentz gas is governed by
the broken Hamiltonian flow
(x˙(t),v˙(t)) = (v(t),0), whenever x(t) Y ,
r
(1.3) ∈
v(t+) = v(t ) 2v(t ) n n , whenever x(t) ∂Y ,
− − x(t) x(t) r
− · ∈
where n denotes the inward unit normal to ∂Y at point x. As can be
x r
easily checked, the measure µ is invariant under the flow (1.3).
r
Let m L (S1) such that
∞
∈
(1.4) m 0 and m(θ)dµ (x,θ) = 1.
r
≥
ZZYr S1
×
The object of interest in the present paper is the distribution of τ
r
under mdµ , ie. the function Φm : R [0,1] defined by
r r + →
(1.5) Φm(t) = mdµ Prob ( (x,v) Y S1 τ (x,v) > t ).
r r − { ∈ r × | r }
More precisely, we are interested in the asymptotic behavior of Φm(t)
r
as r 0+ and for large values of t — large compared to 1/r.
→
FREE PATH LENGTHS FOR PERIODIC LORENTZ GAS 3
As explained in [7] pp. 221–222, knowing the distribution of the
free path lengths in the small r limit has several important implica-
tions. For instance, it leads to the correct asymptotic model for the
Boltzmann-Grad limit of the periodic Lorentz gas — which is not gov-
erned by the linear Boltzmann equation (equation (10) of [13]): see
below. Another application bears on the asymptotic behavior of the
Kolmogorov-Sinai entropy — or equivalently, of the Lyapunov expo-
nent — of the periodic Lorentz gas in the small r limit. We refer to [9]
for a survey of the most recent results and open questions on this sub-
ject, and to [8] which addresses one of these open problems by methods
similar to those developed here.
1.3. Main result. The reference [6] established for each m L (S1)
∞
∈
as in (1.4) the existence of two positive constants C and C such
m m′
that, for each r (0, 1) and each t > 1/r
∈ 2
C C
(1.6) m Φm(t) m′ ;
rt ≤ r ≤ rt
(see Theorems B and C in [6]). In (1.6), the upper bound was proved
by an argument based on Fourier series, while the lower bound was
obtained by a geometric construction exhausting all possible channels,
ie. infinitely long open strips included in Z . The validity of the lower
r
bound in (1.6) was extended to arbitrary space dimensions in [12].
Numerical simulations in [12] suggest the following questions: for
each m L (S1) as in (1.4) does one have, for each t large enough
∞
∈
(say, for each t > 2)
t
(1.7) Φm Λm(t) as ǫ 0?
r r → →
(cid:18) (cid:19)
and, if so, does one have, for some constant C > 0
C
(1.8) Λm(t) as t + ?
∼ t → ∞
We have not been able to fully answer (1.7), but, were (1.7) true,
our main result in this paper (Theorem 1.1 below) would answer (1.8)
by giving an explicit value for C. It confirms the numerical results
obtained in [12] (see figures 4 and 5 there). Throughout the paper, the
notation for the convergence in the sense of Cesaro is as follows:
1 ǫ∗ dr
C limsupf(ǫ) = l means that limsup f(r) = l
− lnǫ r
ǫ→0+ ǫ→0+ | | Zǫ
for some ǫ > 0. (A similar notation is used for the liminf and the lim
∗
in the sense of Cesaro).
4 E. CAGLIOTI ANDF. GOLSE
Theorem 1.1. Let m L (S1) satisfying (1.4). Let t > √2 and let
∞ ∗
∈
t t
Λm(t ) = C limsupΦm ∗ , and Λm(t ) = C liminfΦm ∗ .
+ ∗ − r→0+ r (cid:18)r (cid:19) − ∗ −r→0+ r (cid:18)r (cid:19)
Then
2 2
(1.9) Λm(t ) , and Λm(t ) as t + .
+ ∗ ∼ π2t∗ − ∗ ∼ π2t∗ ∗ → ∞
More precisely,
(1.10) Λm(t ) 2 8kmkL∞ .
∗
− π2t ≤ t 3
(cid:12) ± ∗(cid:12) ∗ −
(cid:12) (cid:12)
A serious shortcom(cid:12)ing of the resu(cid:12)lt above is the need for averaging
(cid:12) (cid:12)
in r before letting r 0+. It seems however that it cannot be avoided,
→
at least by using the techniques of the present paper.
This leads to a natural question, that of the choice of the measure dr
r
to define the Cesaro mean in Theorem 1.1. The reasons for this choice
are made clear by following the proof, but we take this opportunity of
giving an idea of this proof by providing some qualitative argument in
favor of this choice.
The proof of Theorem 1.1 is based on comparing the size r of the
obstacle with the sequence of errors d in the approximation by con-
n
tinued fractions of v /v , ie. of the slope of the linear flow (say, in the
2 1
case where 0 < v < v ). It is naturalin this context to renormalize the
2 1
problem by applying to α = v /v the Gauss map T : x 1 [1] —
2 1 7→ x − x
we refer to the appendix for more details on these notions. Lemma 7.1
below and especially the formula d (α) = αd (T(α)) show that the
n n 1
−
exit time problem with slope α and obstacle of size r is mapped to the
analogous problem with slope Tα and obstacle of size αr. Hence it is
natural to define the Cesaro average in Theorem 1.1 with the measure
dr which is the scale invariant (Haar) measure of the multiplicative
r
group R .
∗+
Remark 1.1. In fact, one can prove that Λm(t ) = Λm(t ) for each
+ ∗ ∗
t large enough (say, for t > 10). (In other words C lim− Φm t∗
∗ ∗ − r→0+ r r
exists). However, this result requires some significant improvements of
(cid:0) (cid:1)
the method used in the present paper; they will be described in [8].
2. A partition of T2
In 1989, R. Thom posed the following problem:
What is the longest orbit of a linear flow with irrational slope
•
on a flat torus with a disk removed?
FREE PATH LENGTHS FOR PERIODIC LORENTZ GAS 5
v
S (v)
r θ
θ
r
Figure 2. The punctured torus Y and the slit S (v)
r r
The answer to this question was found by Blank and Krikorian in [2]
and is summarized as follows. Without loss of generality, assume that
the linear flow is x x + tv where v = (cosθ,sinθ) with θ (0, π).
7→ ∈ 4
The removed disk of diameter r is replaced by a vertical slit S (v) of
r
length r/cosθ with the same center (see figure 2).
Proposition 2.1 (Blank-Krikorian [2] p. 722). Let r (0, 1), θ
∈ 2 ∈
(0, π) and v = (cosθ,sinθ). Assume that tanθ is irrational. There
4
exist three positive numbers l (r,v), l (r,v) and l (r,v) satisfying
A B C
l (r,v) < l (r,v) and l (r,v) = l (r,v)+l (r,v)
A B C A B
and such that, for any orbit γ of the linear flow x x+tv in T2 S (v),
r
7→ \
length(γ) takes one of the three values l (r,v), l (r,v) or l (r,v).
A B C
Conversely, given any l l (r,v),l (r,v),l (r,v) , there exists an
A B C
∈ { }
orbit γ of the flow x x+tv of length l.
7→
Let v be fixed; orbits of length l (r,v) (resp. of length l (r,v),
A B
l (r,v)) are referred to as orbits of type A (resp. of type B, C)1.
C
1Onp. 722of[2],thesentence “Ifthe movingslitfirstmeetsthe fixedslitatthe
bottom,theAandB-orbitsarereversed”mightbethesourceofaslightambiguity
in the definition of the partition above. In the present paper, the orbits of type A
are the shortest, consistently with the table on p. 726 of [2]. Hence the roles of
orbits of type A and B cannot be reversed in the present discussion.
6 E. CAGLIOTI ANDF. GOLSE
0 t t’ t’’
Y (r,v)
B
S (r,v)
B
Y (r,v)
C
S(r,v)
C
Y (r,v)
A
S(r,v)
A
Figure 3. The partition (Y (r,v),Y (r,v),Y (r,v)) of
A B C
T2 S (v)
r
\
Proposition 2.1 defines a partition (Y (r,v),Y (r,v),Y (r,v)) of T2
A B C
\
S (v), where
r
Y (r,v) = x Y S (v) x belongs to an orbit of type A
A r r
{ ∈ \ | }
and Y (r,v) and Y (r,v) are similarly defined. Define further
B C
S (r,v) = y S (v) the v-orbit starting from y is of type A
A r
{ ∈ | }
with analogous definitions for S (r,v) and S (r,v). Clearly Y (r,v)
B C A
(resp. Y (r,v), Y (r,v)) is metrically equivalent to a strip (parallel-
B C
logram) of length l (r,v) and width S (r,v) (resp. of length l (r,v),
A A B
| |
l (r,v) and width S (r,v) , S (r,v) ): see figure 3 below.
C B B
| | | |
Define now λ to be the exit time in the torus with the slit (instead
r
of the disk) removed, ie.
(2.1) λ (z,v) = inf t > 0 z+tv S (v) for each z T2 S (v).
r r r
{ | ∈ } ∈ \
For v = (cosθ,sinθ) with θ (0, π) as above, define
∈ 4
(2.2) ψ (t,v) = dx Prob( z T2 S (v) λ (z,v) t ).
r r r
− { ∈ \ | ≥ }
WiththepartitionofT2 S (v)in(Y (r,v),Y (r,v),Y (r,v)),which is
r A B C
\
metrically equivalent to the disjoint union of three strips as represented
on figure 3, computing ψ (t,v) in terms of the quantities S (r,v) ,
r A
| |
FREE PATH LENGTHS FOR PERIODIC LORENTZ GAS 7
ψ
(t,v)
r
1
slope
−r
slope
−(|S (r,v)|+|S (r,v)|)cosθ
B C
slope
−|S (r,v)|cosθ
C
t
0 l (r,v) l (r,v) l (r,v)
A B C
Figure 4. Graph of t ψ (t,v)
r
7→
l (r,v), S (r,v) , l (r,v), S (r,v) andl (r,v) becomes aneasy task.
A B B C C
| | | |
One finds that
if 0 t l (r,v), then
A
• ≤ ≤
(2.3) ψ (t,v) = 1 tr;
r
−
if l (r,v) t l (r,v), then
A ′ B
• ≤ ≤
ψ (t,v) = 1 l (r,v)r
r ′ A
(2.4) −
(t l (r,v))( S (r,v) + S (r,v) )cosθ;
′ A B C
− − | | | |
if l (r,v) t l (r,v), then
B ′′ C
• ≤ ≤
(2.5) ψ (t ,v) = (l (r,v) t ) S (r,v) cosθ;
r ′′ C ′′ C
− | |
if t l (r,v), then
C
• ≥
(2.6) ψ (t,v) = 0.
r
The graph of t ψ (t,v) is presented on figure 4 below.
r
7→
So far, the distribution ψ (t,v) has been computed in terms of the
r
quantities S (r,v) , l (r,v), S (r,v) , l (r,v), S (r,v) and l (r,v)
A A B B C C
| | | | | |
thatcharacterize thepartition(Y (r,v),Y (r,v),Y (r,v))ofT2 S (v).
A B C r
\
These quantities can be expressed (see [2]) in terms of the continued
fraction expansion of tanθ, in the following manner.
In the following discussion, we freely use the notations recalled in
the Appendix below. Let α = tanθ (0,1); the θ’s for which α Q
∈ ∈
8 E. CAGLIOTI ANDF. GOLSE
forma set of measure 0 and are discarded in the argument below. With
the sequence of errors d in the continued fraction expansion of α, we
n
consider the following partition of the interval (0,1):
(2.7) (0,1) = I , with I = [d ,d ).
n n n n 1
−
n 1
[≥
Each interval I is further partitioned into
n
(2.8)
I = I , with I = [sup(d ,d kd ),d (k 1)d ),
n n,k n,k n n 1 n n 1 n
− − − − −
1≤[k≤an
so that eventually we arrive at the following nested partition of (0,1):
(2.9) (0,1) = I .
n,k
n[≥11≤[k≤an
Proposition 2.2 (Blank-Krikorian [2] p. 726). Let θ (0, π) and
∈ 4
r (0, 1). Let v = (cosθ,sinθ) and set α = tanθ (it is assumed that
∈ 2
α / Q). For R = r , the integers n 1 and k such that 1 k a
∈ cosθ ≥ ≤ ≤ n
(a being the n-th term in the continued fraction expansion of α) are
n
definedby the factthat R I (sincethe intervals I form a partition
n,k n,k
∈
of (0,1): see (2.9)). Then
l (r,v) = q and S (r,v) = R d ,
A n A n
• | | −
l (r,v) = q +kq and S (r,v) = R (d kd ),
B n 1 n B n 1 n
• − | | − − −
l (r,v) = q +(k+1)q and S (r,v) = d (k 1)d R.
C n 1 n C n 1 n
• − | | − − − −
Using these values, we arrive at the following expression for ψ (t,v),
r
whenever R = r I (see figure 5 below):
cosθ ∈ n,k
if 0 tcosθ q , then
n
• ≤ ≤
(2.10) ψ (t,v) = 1 tr,
r
−
if q tcosθ q +kq , then
n n 1 n
• ≤ ≤ −
(2.11) ψ (t,v) = 1 Rq d (tcosθ q ),
r n n n
− − −
if q +kq tcosθ q +(k +1)q , then
n 1 n n 1 n
• − ≤ ≤ −
ψ (t,v) = 1 Rq d [q +(k 1)q ]
r n n n 1 n
(2.12) − − − −
[R (d (k 1)d )](tcosθ q kq ),
n 1 n n 1 n
− − − − − − − −
if tcosθ q +(k +1)q , then
n 1 n
• ≥ −
(2.13) ψ (t,v) = 0.
r
Finally, the discussion above leads to the statistics of the exit time
λ (z,v) defined in (2.1) corresponding to the torus T2 with the slit
r
S (v) removed. In the case of the torus with the disk removed, the
r
FREE PATH LENGTHS FOR PERIODIC LORENTZ GAS 9
ψ
r(t,v)
If R is in I
n,k
1
slope
−r
slope
−dncosθ
slope
Γ r−[dn−1−(k−1)dn]cosθ
0 qn qn−1+kqn qn−1+(k+1)qn tcosθ
Figure 5. Graph of t ψ (t,v) for R I
r n,k
7→ ∈
corresponding exit time τ (x,v) defined in (1.2) is related to λ (z,v)
r r
by the obvious inequalities
(2.14) λ (x,v) r τ (x,v) λ (x,v)+ r for each x Y S (v).
r − 2 ≤ r ≤ r 2 ∈ r \ r
Define
(2.15) φ (t,v) = dx Prob( x Y S (v) τ (x,v) t );
r r r r
− { ∈ \ | ≥ }
because of (2.14), one has
(2.16) ψ (t+ r,v) φ (t,v) ψ (t r,v), t r .
r 2 ≤ r ≤ r − 2 ≥ 2
The remaining part of the paper uses the evaluation of φ based on this
r
inequality together with the formulas (2.10), (2.11), (2.12) and (2.13)
for ψ .
r
3. An ergodic theorem
Given α (0,1) and ǫ (0,1), we define
∈ ∈
(3.1) N(α,ǫ) = inf n N d (α) < ǫ .
n+1
{ ∈ | }
In terms of the partition (2.7) of (0,1), N(α,ǫ) can be equivalently
defined by the condition
(3.2) ǫ I .
N(α,ǫ)+1
∈
We start by recalling the following more or less classical lemma.
10 E. CAGLIOTI ANDF. GOLSE
Lemma 3.1. For ae. α (0,1), one has
∈
N(α,ǫ) 12ln2 lnǫ as ǫ 0+.
∼ − π2 →
Proof. The definition of N(α,ǫ) and the third formula in Lemma 7.1
imply that
N(α,ǫ) 1 N(α,ǫ)
−
(3.3) lnTjα lnǫ < lnTjα,
− ≤ − −
j=0 j=0
X X
where T is the Gauss map (7.1). First we prove that N(α,ǫ) +
→ ∞
as ǫ 0+ ae. in α (0,1). Indeed, let C > 0 and let
→ ∈
E = α (0,1) N(α,ǫ) C, for all ǫ (0, 1) .
C { ∈ | ≤ ∈ 2 }
For all α E and all ǫ (0, 1)
∈ C ∈ 2
C
lnǫ lnTjα;
− ≤ −
j=0
X
therefore, for all ǫ (0, 1),
∈ 2
1
lnǫ dg meas (E ) (C +1) ( lnα)dg(α) < + ,
C
− · − ≤ − ∞
Z0
since the measure dg in (7.2) is invariant under T. This implies that
(3.4) for each C > 0, dg meas (E ) = 0.
C
−
Because N(α,ǫ) is a nonincreasing function of ǫ,
c
(3.5) for each α E ,N(α,ǫ) +
m
∈ → ∞
!
m 1
[≥
as ǫ 0+ and
→
(3.6) dg meas E = 0.
m
−
!
m 1
[≥
Secondly, by Birkhoff’s ergodic theorem, there exists a dg-negligeable
set E such that
′
1 N−1 1
for each α (E )c, lnTjα ( lnα)dg(α)
′
∈ N − → −
0 Z0
X
as N + , since the Gauss transformation T of (0,1) is ergodic with
→ ∞
respect to the invariant measure dg(α) — see the Appendix below.
