The distribution of free path lengths in the periodic Lorentz gas

The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems

Jens Marklof  and  Andreas Strömbergsson School of Mathematics, University of Bristol, Bristol BS8 1TW, U.K.
   j.marklof@bristol.ac.uk
Department of Mathematics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden
   astrombe@math.kth.se
   Present address: Dept. of Mathematics, Box 480, Uppsala University, SE-75106 Uppsala, Sweden
   astrombe@math.uu.se
29 June 2007, revised 14 January 2008, to appear in the Annals of Mathematics
Abstract.

The periodic Lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers, and is one of the fundamental models for chaotic diffusion. In the present paper we investigate the Boltzmann-Grad limit, where the radius of each scatterer tends to zero, and prove the existence of a limiting distribution for the free path length. We also discuss related problems, such as the statistical distribution of directions of lattice points that are visible from a fixed position.

J.M. has been supported by an EPSRC Advanced Research Fellowship and a Philip Leverhulme Prize. A.S. is a Royal Swedish Academy of Sciences Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation.

1. Introduction

1.1. The periodic Lorentz gas

The Lorentz gas, originally introduced by Lorentz [20] in 1905 to model the motion of electrons in a metal, describes an ensemble of non-interacting point particles in an infinite array of spherical scatterers. Lorentz was in particular interested in the stochastic properties of the dynamics that emerge in the Boltzmann-Grad limit, where the radius of each scatterer tends to zero.

In the present and subsequent papers [23], [24], [25] we investigate the periodic set-up, where the scatterers are placed at the vertices of a euclidean lattice (Figure 1). We will identify a new random process that governs the macroscopic dynamics of a particle cloud in the Boltzmann-Grad limit. In the case of a Poisson-distributed (rather than periodic) configuration of scatterers, the limiting process is described by the linear Boltzmann equation, see Gallavotti [16], Spohn [36], and Boldrighini, Bunimovich and Sinai [8]. It already follows from the estimates in [9], [19] that the linear Boltzmann equation does not hold in the periodic set-up; this was pointed out recently by Golse [18].

The first step towards the proof of the existence of a limiting process for the periodic Lorentz gas is the understanding of the distribution of the free path length in the limit , which is the key result of the present paper. The distribution of the free path lengths in the periodic Lorentz gas was already investigated by Polya, who rephrased the problem in terms of the visibility in a (periodic) forest [29]. We complete the analysis of the limiting process in [23], [24] and [25], where we establish a Markov property, and provide explicit formulas and asymptotic estimates for the limiting distributions.

Our results complement classical studies in ergodic theory, where one is interested in the stochastic properties in the limit of long times, with the radius of each scatterer being fixed. Here Bunimovich and Sinai [10] proved, in the case of a finite horizon and in dimension , that the dynamics is diffusive in the limit of large times, and satisfies a central limit theorem. “Finite horizon” means that the scatterers are sufficiently large so that the path length between consecutive collisions is bounded; this hypothesis was recently removed by Szasz and Varju [31] after initial work by Bleher [2]. For related recent studies of statistical properties of the two-dimensional periodic Lorentz gas, see also [13], [27], [28]. There is at present no proof of the central limit theorem for higher dimensions, even in the case of finite horizon [12], [1].

fixed

Figure 1. Left: The periodic Lorentz gas in “microscopic” coordinates—the lattice remains fixed as the radius of the scatterer tends to zero. Right: The 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.

Since the point particles of the Lorentz gas are non-interacting, we can reduce the problem to the study of the billiard flow

(1.1)

where is the complement of the set (the “billiard domain”), and is its unit tangent bundle (the “phase space”). denotes the open ball of radius , centered at the origin. A point in is parametrized by , with denoting the position and the velocity of the particle. The Liouville measure of is

(1.2)

where and refer to the Lebesgue measures on (restricted to ) and , respectively.

The free path length for the initial condition is defined as

(1.3)

That is, is the first time at which a particle with initial data hits a scatterer.

From now on we will assume, without loss of generality, that has covolume one.

Theorem 1.1.

Fix a lattice of covolume one, let , and let be a Borel probability measure on absolutely continuous with respect to Lebesgue measure.111The condition ensures that is defined for sufficiently small. In Section 4 we also consider variants of Theorem 1.1 where the initial position is near , e.g., . Then there exists a continuous probability density on such that, for every ,

(1.4)

The limiting density is in fact “universal” for generic , i.e.,

(1.5)

is independent of and , for Lebesgue-almost every . Theorem 1.1 is proved in Section 4, it is closely related to the lattice point problem studied in Section 3. The asymptotic tails of the limiting distribution are calculated in [25]. In Section 4 we generalize Theorem 1.1 in several ways. We consider for instance the distribution of free paths that hit a given point on the scatterer, which will be crucial in the characterization of the limiting random process in [23].

Theorem 1.1 shows that the free path length scales like . This suggests to re-define position and time and use the “macroscopic” coordinates

(1.6)

We now state a macroscopic version of Theorem 1.1, which is a corollary of the proof of Theorem 1.1 (see Section 9.2). Here

(1.7)

is the corresponding macroscopic free path length.

Theorem 1.2.

Fix a lattice of covolume one and let be a Borel probability measure on absolutely continuous with respect to Lebesgue measure. Then, for every ,

(1.8)

with as in (1.5).

Variants of Theorem 1.2 were recently established by Boca and Zaharescu [7] in dimension , using methods from analytic number theory; cf. also their earlier work with Gologan [4], and the paper by Calglioti and Golse [11]. Our approach uses dynamics and equidistribution of flows on homogeneous spaces (the details are developed in Section 5), and works in arbitrary dimension. Previous work in higher dimension includes the papers by Bourgain, Golse and Wennberg [9], [19] who provide tail estimates of possible limiting distributions of converging subsequences. More details on the existing literature can be found in the survey [17].

1.2. Related lattice point problems

fixed

Figure 2. Left: How many lattice balls of radius does a random ray of length intersect? Right: What are the statistical properties of the directions of the affine lattice points inside a large ball?

The key to the understanding of the Boltzmann-Grad limit of the periodic Lorentz gas are lattice point problems for thinly stretched domains, which are randomly rotated or sheared. In Sections 2 and 3 we discuss two problems of independent interest that fall into this category: the distribution of spheres that intersect a randomly directed ray, and the statistical properties of the directions of lattice points (Figure 2). Section 6 discusses the general class of problems of this type.

Let us for example consider the affine lattice , with the observer located at the origin. The directions of all lattice points with distance are represented by points on the unit circle,

(1.9)

We identify the circle with the unit interval via the map , and therefore the distribution of directions is reformulated as a problem of distribution mod 1 of the numbers

(1.10)

We label these numbers in order by

(1.11)

and define in addition . It is not hard to see that this sequence (or rather: this sequence of sequences) is uniformly distributed mod 1, i.e., for every ,

(1.12)

This (classical) equidistribution statement follows from the fact that the asymptotic number of lattice points in a fixed sector of a large disc is proportional to the volume of the sector.

A popular way to characterize the “randomness” of a uniformly distributed sequence is the statistics of gaps. The following theorem, which is a corollary of more general results in Section 2, shows that there is a limiting gap distribution when .

Theorem 1.3.

For every there exists a distribution function on (continuous except possibly at ) such that for every ,

(1.13)

We will provide explicit formulas for , which clearly deviate from the statistics of independent random variables from a Poisson process, where . It is remarkable that, for , the limiting distribution is independent of and coincides with the gap distribution for the fractional parts of calculated by Elkies and McMullen [14]; cf. Figure 3. There is a deep reason for this apparent coincidence, which we will return to in the next section.

The statistics are different for . In particular has a jump discontinuity at for every , which exactly accounts for the multiplicities in the sequence (1.11); removing all repetitions from that sequence results in a limiting gap distribution which is continuous on all , see Corollary 2.7 below. In the particular case this recovers a result of Boca, Cobeli and Zaharescu [3], which is closely related to the statistical distribution of Farey fractions (see also Boca and Zaharescu [5]).

The only previously known result for non-zero values of is by Boca and Zaharescu [6], who calculated the limit of the pair correlation function on average over . (The pair correlation function is essentially the variance of the probability studied in Section 2.) Contrary to the behaviour of the gap probability , the limiting pair correlation function is the same as for random variables from a Poisson process.222Boca and Zaharescu consider a slightly different sequence of directions, which is obtained by replacing the last condition in (1.10) with . This sequence is however not uniformly distributed modulo one, which explains the discrepancy with the Poisson pair correlation function observed in [6].

Figure 3. Left: The distribution of gaps in the sequence , , vs. the Elkies-McMullen distribution. Right: Gap distribution for the directions of the vectors with , , . The continuous curve is the Elkies-McMullen distribution.

1.3. Outline of the paper

Sections 24 give a detailed account of the main results of this paper. Section 2 discusses the statistical properties of affine lattice points inside a large sphere that are projected onto the unit sphere. A dual problem is the question of the probability that a ray of length pointing in a random direction intersects exactly lattice spheres whose radius scales as . The solution of the latter problem is provided in Section 3, and applied in Section 4 to the distribution of the free path lengths of the Lorentz gas. Both of the above lattice point problems fall into a general class of lattice point problems in randomly sheared or rotated domains, which are discussed in Section 6. The central idea for the solution of such questions is to exploit equidistribution results for flows on the homogeneous spaces and , which represent the space of lattices (resp. affine lattices) of covolume one. We establish the required ergodic-theoretic results in Section 5. The key ingredient is Ratner’s theorem [30] on the classification of ergodic measures invariant under a unipotent flow. We provide useful integration formulas on and in Section 7 and in Section 8 we apply these to our limit functions. Detailed proofs of the main limit theorems in Sections 24 are given in Section 9. The proofs for Section 2 are virtually identical to those of the corresponding theorems in Section 3.

2. Distribution of visible lattice points

2.1. Lattices

Let be a euclidean lattice of covolume one. Recall that for some and that therefore the homogeneous space parametrizes the space of lattices of covolume one.

Let be the semidirect product group with multiplication law

(2.1)

An action of on can be defined as

(2.2)

Each affine lattice (i.e. translate of a lattice) of covolume one in can then be expressed as for some , and the space of affine lattices is then represented by where . We denote by and the Haar measure on and , respectively, normalized in such a way that they represent probability measures on and .

If , say for , , we see that

(2.3)

for all

(2.4)

the principal congruence subgroup. This means that the space of affine lattices with can be parametrized by the homogeneous space (this is not necessarily one-to-one). We denote by the Haar measure on which is normalized as a probability measure on .

2.2. Basic set-up

We fix a lattice of covolume one, and fix, once and for all, a choice of such that . Given we then define the affine lattice

(2.5)

Consider the set of lattice points inside the ball of radius , or, more generally, the spherical shell

(2.6)

For large there are asymptotically such points, where is the volume of the unit ball. For each , we study the corresponding directions,

(2.7)

where denotes the -sphere of radius . It is well known that, as , these points become uniformly distributed on : For any set with boundary of measure zero (with respect to the volume element on ) we have

(2.8)

Recall that .

2.3. Distribution in small discs

We are interested in the fine-scale distribution of the directions to points in , e.g., in the probability of finding directions in a small disc with random center . We define to be the open disc with center and volume

(2.9)

The radius of is thus (if ). We introduce the counting function

(2.10)

for the number of points in . The motivation for the definition (2.9) is that it implies, via (2.8), that the expectation value for the counting function is asymptotically equal to (for and fixed):

(2.11)

where is any probability measure on with continuous density, and is the disc centered at .

Theorem 2.1.

Let be a Borel probability measure on absolutely continuous with respect to Lebesgue measure. Then, for every and , the limit

(2.12)

exists, and for fixed the convergence is uniform with respect to in any compact subset of . The limit function is given by

(2.13)

where

(2.14)
(2.15)

In particular, is continuous in and independent of and .

In the above, we use the notation . Although the use of is superfluous at this point (since does not contain zero), it appears as the natural object in the proof. This subtlety is due to the fact that for generic we have but .

Theorem 2.1 says that the limiting distribution is given by the probability that there are points of a random lattice in the cone , and for is the corresponding probability for a random affine lattice. Hence in particular is independent of when .

Remark 2.2.

We will furthermore prove that when the function is with respect to ; see Section 8.5. We expect that the same statement should also be true for any fixed .

Remark 2.3.

In the case , and our distribution coincides with Elkies and McMullen’s limiting distribution [14] for the probability of finding elements of the sequence mod 1 () in a randomly shifted interval of length (). Although the two problems are seemingly unrelated, the reason for this coincidence is that both results use equidistribution of translates of different orbits on the space of affine lattices with respect to the same test functions.

Remark 2.4.

By a general statistical argument, cf. e.g. [14], [22], Theorem 1.3 is an immediate corollary of Theorem 2.1 in the case , , with the limit function explicitly given by

(2.16)

The continuity of for follows from Remark 2.2.

To exhibit explicitly the group action which will play a central role in the proof of the above statements, it is convenient to realize as the homogeneous space by setting with and . The stabilizer of is isomorphic to (acting from the right), where is identified with the subgroup

(2.17)

Then

(2.18)

and

(2.19)

is the number of points in . Note that is left-invariant under the action of and thus may be viewed as a function on . The statement equivalent to Theorem 2.1 is now that, if is a Borel probability measure on absolutely continuous with respect to Haar measure, then

(2.20)

2.4. Visible lattice points

In the study of directions of affine lattice points it is natural to restrict our attention to those points that are visible from the origin. That is, we consider the set of directions without counting multiplicities. Non-trivial multiplicities are only obtained when the -linear span of and the components of has dimension . If then the multiplicities are statistically insignificant; in fact they can only occur along at most a single line through the origin, and thus restricting to considering only the visible lattice points still yields the same limit distribution as in Theorem 2.1.

Hence from now on we will assume . If then the visible lattice points are exactly the primitive lattice points, i.e. those points for which , . In the general case (, ), the set of visible lattice points is:

(2.21)

From now on in this section we will assume that is the minimal integer which gives . Given we set ; then by a sieving argument using (2.21) and (2.8) one shows that for any set with boundary of measure zero,

(2.22)

When this specializes to the well-known fact that the asymptotic density of the primitive points in is . It follows from (2.22) that if we introduce the following analogue of (2.10) for visible lattice points:

(2.23)

then the expectation value for is again asymptotically equal to :

(2.24)

for any fixed , and as in (2.11).

Theorem 2.5.

Let be a Borel probability measure on absolutely continuous with respect to Lebesgue measure. Then, for every and , the limit

(2.25)

exists, and for fixed the convergence is uniform with respect to in any compact subset of . The limit function is given by

(2.26)

In particular, is continuous in and independent of and .

Remark 2.6.

The function is with respect to . This is proved by adapting the arguments of Sections 7.1 and 8.5 to the setting of visible lattice points.

In dimension , considering only visible lattice points gives a variant of Theorem 1.3 with an everywhere continuous distribution function: Take , and consider the set of rescaled directions

(2.27)

Let us label these numbers in order by

(2.28)

and define in addition . Note that this is exactly the sequence which is obtained from (1.11) by removing all repetitions. We now have:

Corollary 2.7.

There exists a distribution function on , continuous on all of , such that for every ,

(2.29)
Proof.

Just as in Remark 2.4, the limit relation (2.29) follows from Theorem 2.5 together with the fact as (cf. (2.22)), and is explicitly given by

(2.30)

Note that for all , since is star shaped. Hence

(2.31)

The continuity of for follows from Remark 2.2, or Remark 2.6. Furthermore, in Section 8.5 we will prove that (for ),

(2.32)

and this implies that is also continuous at . ∎

When , Corollary 2.7 specializes to give the limiting gap distribution for directions of primitive lattice points in , which was proved earlier by Boca, Cobeli and Zaharescu [3].

The proofs of Theorems 2.1 and 2.5 are virtually identical to those of Theorems 3.1 and 3.7; we will therefore only outline the differences in Section 9.4. In [26] we carry out a more detailed statistical analysis of the distribution of visible lattice points, which yields generalizations of Theorems 2.1 and 2.5, and also provide explicit formulas and tail estimates of the limiting distributions.

3. The number of spheres in a random direction

We now turn to a lattice point problem that is in some sense dual to the one studied in the previous Section 2. Its solution will also answer the question of the distribution of free path lengths in the periodic Lorentz gas, see Section 4 below for details.

3.1. Spheres centered at lattice points

We place at each lattice point a ball of small radius and consider the set . The set of balls with centers inside the shell (2.6) is

(3.1)

Note that we remove any ball at (this is only relevant in the case ). Furthermore we will always keep , so that lies outside each of the balls in our set. We are interested in the number of intersections of this set with a ray starting at the origin that points in the random direction distributed according to the probability measure . That is

(3.2)

If , then a ray in direction hits the ball if and only if

(3.3)

with the disc

(3.4)

We will again use the shorthand . The radius of this disc is , for . Hence the number of balls hit by a ray in direction is

(3.5)

compare (2.19).

For any as in (2.11), one finds for the expectation value as ,

(3.6)