on the Periodic Lorentz Gas
The Drude-Lorentz model for the motion of electrons in a solid is a classical model in statistical mechanics, where electrons are represented as point particles bouncing on a fixed system of obstacles (the atoms in the solid). Under some appropriate scaling assumption — known as the Boltzmann-Grad scaling by analogy with the kinetic theory of rarefied gases — this system can be described in some limit by a linear Boltzmann equation, assuming that the configuration of obstacles is random [G. Gallavotti, [Phys. Rev. (2) 185 (1969), 308]). The case of a periodic configuration of obstacles (like atoms in a crystal) leads to a completely different limiting dynamics. These lecture notes review several results on this problem obtained in the past decade as joint work with J. Bourgain, E. Caglioti and B. Wennberg.
Key words and phrases:Periodic Lorentz gas, Boltzmann-Grad limit, Linear Boltzmann equation, Mean free path, Distribution of free path lengths, Continued fractions, Farey fractions
2000 Mathematics Subject Classification:82C70, 35B27 (82C40, 11A55, 11B57, 11K50)
- 1 The Lorentz kinetic theory for electrons
- 2 The Lorentz gas in the Boltzmann-Grad limit with a Poisson distribution of obstacles
- 3 Santaló’s formula for the geometric mean free path
- 4 Estimates for the distribution of free-path lengths
- 5 A negative result for the Boltzmann-Grad limit of the periodic Lorentz gas
- 6 Coding particle trajectories with continued fractions
- 7 An ergodic theorem for collision patterns
- 8 Explicit computation of the transition probability
- 9 A kinetic theory in extended phase-space for the Boltzmann-Grad limit of the periodic Lorentz gas
Introduction: from particle dynamics to kinetic models
The kinetic theory of gases was proposed by J. Clerk Maxwell [34, 35] and L. Boltzmann  in the second half of the XIXth century. Because the existence of atoms, on which kinetic theory rested, remained controversial for some time, it was not until many years later, in the XXth century, that the tools of kinetic theory became of common use in various branches of physics such as neutron transport, radiative transfer, plasma and semiconductor physics…
Besides, the arguments which Maxwell and Boltzmann used in writing what is now known as the “Boltzmann collision integral” were far from rigorous — at least from the mathematical viewpoint. As a matter of fact, the Boltzmann equation itself was studied by some of the most distinguished mathematicians of the XXth century — such as Hilbert and Carleman — before there were any serious attempt at deriving this equation from first principles (i.e. molecular dynamics.) Whether the Boltzmann equation itself was viewed as a fundamental equation of gas dynamics, or as some approximate equation valid in some well identified limit is not very clear in the first works on the subject — including Maxwell’s and Boltzmann’s.
It seems that the first systematic discussion of the validity of the Boltzmann equation viewed as some limit of molecular dynamics — i.e. the free motion of a large number of small balls subject to binary, short range interaction, for instance elastic collisions — goes back to the work of H. Grad . In 1975, O.E. Lanford gave the first rigorous derivation  of the Boltzmann equation from molecular dynamics — his result proved the validity of the Boltzmann equation for a very short time of the order of a fraction of the reciprocal collision frequency. (One should also mention an earlier, “formal derivation” by C. Cercignani  of the Boltzmann equation for a hard sphere gas, which considerably clarified the mathematical formulation of the problem.) Shortly after Lanford’s derivation of the Boltzmann equation, R. Illner and M. Pulvirenti managed to extend the validity of his result for all positive times, for initial data corresponding with a very rarefied cloud of gas molecules .
An important assumption made in Boltzmann’s attempt at justifying the equation bearing his name is the “Stosszahlansatz”, to the effect that particle pairs just about to collide are uncorrelated. Lanford’s argument indirectly established the validity of Boltzmann’s assumption, at least on very short time intervals.
In applications of kinetic theory other than rarefied gas dynamics, one may face the situation where the analogue of the Boltzmann equation for monatomic gases is linear, instead of quadratic. The linear Boltzmann equation is encountered for instance in neutron transport, or in some models in radiative transfer. It usually describes a situation where particles interact with some background medium — such as neutrons with the atoms of some fissile material, or photons subject to scattering processes (Rayleigh or Thomson scattering) in a gas or a plasma.
In some situations leading to a linear Boltzmann equation, one has to think of two families of particles: the moving particles whose phase space density satisfies the linear Boltzmann equation, and the background medium that can be viewed as a family of fixed particles of a different type. For instance, one can think of the moving particles as being light particles, whereas the fixed particles can be viewed as infinitely heavier, and therefore unaffected by elastic collisions with the light particles. Before Lanford’s fundamental paper, an important — unfortunately unpublished — preprint by G. Gallavotti  provided a rigorous derivation of the linear Boltzmann equation assuming that the background medium consists of fixed, independent like hard spheres whose centers are distributed in the Euclidian space under Poisson’s law. Gallavotti’s argument already possessed some of the most remarkable features in Lanford’s proof, and therefore must be regarded as an essential step in the understanding of kinetic theory.
However, Boltzmann’s Stosszahlansatz becomes questionable in this kind of situation involving light and heavy particles, as potential correlations among heavy particles may influence the light particle dynamics. Gallavotti’s assumption of a background medium consisting of independent hard spheres excluded this this possibility. Yet, strongly correlated background media are equally natural, and should also be considered.
The periodic Lorentz gas discussed in these notes is one example of this type of situation. Assuming that heavy particles are located at the vertices of some lattice in the Euclidian space clearly introduces about the maximum amount of correlation between these heavy particles. This periodicity assumption entails a dramatic change in the structure of the equation that one obtains under the same scaling limit that would otherwise lead to a linear Boltzmann equation.
Therefore, studying the periodic Lorentz gas can be viewed as one way of testing the limits of the classical concepts of the kinetic theory of gases.
Most of the material presented in these lectures is the result of collaboration with several authors: J. Bourgain, E. Caglioti, H.S. Dumas, L. Dumas and B. Wennberg, whom I wish to thank for sharing my interest for this problem. I am also grateful to C. Boldighrini and G. Gallavotti for illuminating discussions on this subject.
1. The Lorentz kinetic theory for electrons
In the early 1900’s, P. Drude  and H. Lorentz  independently proposed to describe the motion of electrons in metals by the methods of kinetic theory. One should keep in mind that the kinetic theory of gases was by then a relatively new subject: the Boltzmann equation for monatomic gases appeared for the first time in the papers of J. Clerk Maxwell  and L. Boltzmann . Likewise, the existence of electrons had been established shortly before, in 1897 by J.J. Thomson.
The basic assumptions made by H. Lorentz in his paper  can be summarized as follows.
First, the population of electrons is thought of as a gas of point particles described by its phase-space density , that is the density of electrons at the position with velocity at time .
Electron-electron collisions are neglected in the physical regime considered in the Lorentz kinetic model — on the contrary, in the classical kinetic theory of gases, collisions between molecules are important as they account for momentum and heat transfer.
However, the Lorentz kinetic theory takes into account collisions between electrons and the surrounding metallic atoms. These collisions are viewed as simple, elastic hard sphere collisions.
Since electron-electron collisions are neglected in the Lorentz model, the equation governing the electron phase-space density is linear. This is at variance with the classical Boltzmann equation, which is quadratic because only binary collisions involving pairs of molecules are considered in the kinetic theory of gases.
With the simple assumptions above, H. Lorentz arrived at the following equation for the phase-space density of electrons :
In this equation, is the Lorentz collision integral, which acts on the only variable in the phase-space density . In other words, for each continuous function , one has
and the notation
The other parameters involved in the Lorentz equation are the mass of the electron, and , respectively the density and radius of metallic atoms. The vector field is the electric force. In the Lorentz model, the self-consistent electric force — i.e. the electric force created by the electrons themselves — is neglected, so that take into account the only effect of an applied electric field (if any). Roughly speaking, the self consistent electric field is linear in , so that its contribution to the term would be quadratic in , as would be any collision integral accounting for electron-electron collisions. Therefore, neglecting electron-electron collisions and the self-consistent electric field are both in accordance with assuming that .
The line of reasoning used by H. Lorentz to arrive at the kinetic equations above is based on the postulate that the motion of electrons in a metal can be adequately represented by a simple mechanical model — a collisionless gas of point particles bouncing on a system of fixed, large spherical obstacles that represent the metallic atoms. Even with the considerable simplification in this model, the argument sketched in the article  is little more than a formal analogy with Boltzmann’s derivation of the equation now bearing his name.
This suggests the mathematical problem, of deriving the Lorentz kinetic equation from a microscopic, purely mechanical particle model. Thus, we consider a gas of point particles (the electrons) moving in a system of fixed spherical obstacles (the metallic atoms). We assume that collisions between the electrons and the metallic atoms are perfectly elastic, so that, upon colliding with an obstacle, each point particle is specularly reflected on the surface of that obstacle.
Undoubtedly, the most interesting part of the Lorentz kinetic equation is the collision integral which does not seem to involve . Therefore we henceforth assume for the sake of simplicity that there is no applied electric field, so that
In that case, electrons are not accelerated between successive collisions with the metallic atoms, so that the microscopic model to be considered is a simple, dispersing billiard system — also called a Sinai billiard. In that model, electrons are point particles moving at a constant speed along rectilinear trajectories in a system of fixed spherical obstacles, and specularly reflected at the surface of the obstacles.
More than 100 years have elapsed since this simple mechanical model was proposed by P. Drude and H. Lorentz, and today we know that the motion of electrons in a metal is a much more complicated physical phenomenon whose description involves quantum effects.
Yet the Lorentz gas is an important object of study in nonequilibrium satistical mechanics, and there is a very significant amount of literature on that topic — see for instance  and the references therein.
The first rigorous derivation of the Lorentz kinetic equation is due to G. Gallavotti [18, 19], who derived it from from a billiard system consisting of randomly (Poisson) distributed obstacles, possibly overlapping, considered in some scaling limit — the Boltzmann-Grad limit, whose definition will be given (and discussed) below. Slightly more general, random distributions of obstacles were later considered by H. Spohn in .
While Gallavotti’s theorem bears on the convergence of the mean electron density (averaging over obstacle configurations), C. Boldrighini, L. Bunimovich and Ya. Sinai  later succeeded in proving the almost sure convergence (i.e. for a.e. obstacle configuration) of the electron density to the solution of the Lorentz kinetic equation.
In any case, none of the results above says anything on the case of a periodic distribution of obstacles. As we shall see, the periodic case is of a completely different nature — and leads to a very different limiting equation, involving a phase-space different from the one considered by H. Lorentz — i.e. — on which the Lorentz kinetic equation is posed.
The periodic Lorentz gas is at the origin of many challenging mathematical problems. For instance, in the late 1970s, L. Bunimovich and Ya. Sinai studied the periodic Lorentz gas in a scaling limit different from the Boltzmann-Grad limit studied in the present paper. In , they showed that the classical Brownian motion is the limiting dynamics of the Lorentz gas under that scaling assumption — their work was later extended with N. Chernov: see . This result is indeed a major achievement in nonequilibrium statistical mechanics, as it provides an example of an irreversible dynamics (the heat equation associated with the classical Brownian motion) that is derived from a reversible one (the Lorentz gas dynamics).
2. The Lorentz gas in the Boltzmann-Grad limit
with a Poisson distribution of obstacles
Before discussing the Boltzmann-Grad limit of the periodic Lorentz gas, we first give a brief description of Gallavotti’s result [18, 19] for the case of a Poisson distribution of independent, and therefore possibly overlapping obstacles. As we shall see, Gallavotti’s argument is in some sense fairly elementary, and yet brilliant.
First we define the notion of a Poisson distribution of obstacles. Henceforth, for the sake of simplicity, we assume a -dimensional setting.
The obstacles (metallic atoms) are disks of radius in the Euclidian plane , centered at . Henceforth, we denote by
We further assume that the configurations of obstacle centers are distributed under Poisson’s law with parameter , meaning that
where denotes the surface, i.e. the -dimensional Lebesgue measure of a measurable subset of the Euclidian plane .
This prescription defines a probability on countable subsets of the Euclidian plane .
Obstacles may overlap: in other words, configurations such that
are not excluded. Indeed, excluding overlapping obstacles means rejecting obstacles configurations such that for some . In other words, is replaced with
(where is a normalizing coefficient.) Since the term
the obstacles are no longer independent under this new probability measure.
Next we define the billiard flow in a given obstacle configuration . This definition is self-evident, and we give it for the sake of completeness, as well as in order to introduce the notation.
Given a countable subset of the Euclidian plane , the billiard flow in the system of obstacles defined by is the family of mappings
defined by the following prescription.
Whenever the position of a particle lies outside the surface of any obstacle, that particle moves at unit speed along a rectilinear path:
and, in case of a collision with the -th obstacle, is specularly reflected on the surface of that obstacle at the point of impingement, meaning that
where denotes the reflection with respect to the line :
Then, given an initial probability density on the single-particle phase-space with support outside the system of obstacles defined by , we define its evolution under the billiard flow by the formula
Let be the sequence of collision times for a particle starting from in the direction at in the configuration of obstacles : in other words,
Denoting and , the evolved single-particle density is a.e. defined by the formula
In the case of physically admissible initial data, there should be no particle located inside an obstacle. Hence we assumed that in the union of all the disks of radius centered at the . By construction, this condition is obviously preserved by the billiard flow, so that also vanishes whenever belongs to a disk of radius centered at any .
As we shall see shortly, when dealing with bounded initial data, this constraint disappears in the (yet undefined) Boltzmann-Grad limit, as the volume fraction occupied by the obstacles vanishes in that limit.
Therefore, we shall henceforth neglect this difficulty and proceed as if were any bounded probability density on .
Our goal is to average the summation above in the obstacle configuration under the Poisson distribution, and to identify a scaling on the obstacle radius and the parameter of the Poisson distribution leading to a nontrivial limit.
The parameter has the following important physical interpretation. The expected number of obstacle centers to be found in any measurable subset of the Euclidian plane is
The average of the first term in the summation defining is
(where denotes the mathematical expectation) since the condition means that the tube of width and length contains no obstacle center.
Henceforth, we seek a scaling limit corresponding to small obstacles, i.e. and a large number of obstacles per unit volume, i.e. .
There are obviously many possible scalings satisfying this requirement. Among all these scalings, the Boltzmann-Grad scaling in space dimension is defined by the requirement that the average over obstacle configurations of the first term in the series expansion for the particle density has a nontrivial limit.
In order for the average of the first term above to have a nontrivial limit, one must have
Under this assumption
Gallavotti’s idea is that this first term corresponds with the solution at time of the equation
that involves only the loss part in the Lorentz collision integral, and that the (average over obstacle configuration of the) subsequent terms in the sum defining the particle density should converge to the Duhamel formula for the Lorentz kinetic equation.
After this necessary preliminaries, we can state Gallavotti’s theorem.
Theorem 2.1 (Gallavotti ).
Let be a continuous, bounded probability density on , and let
where is the billiard flow in the system of disks of radius centered at the elements of . Assuming that the obstacle centers are distributed under the Poisson law of parameter with , the expected single particle density
uniformly on compact -sets, where is the solution of the Lorentz kinetic equation
where denotes the rotation of an angle .
End of the proof of Gallavotti’s theorem.
The general term in the summation giving is
and its average under the Poisson distribution on is
where is the tube of width around the particle trajectory colliding first with the obstacle centered at , …, and whose -th collision is with the obstacle centered at .
As before, the surface of that tube is
In the -th term, change variables by expressing the positions of the encountered obstacles in terms of free flight times and deflection angles:
The volume element in the -th integral is changed into
The measure in the left-hand side is invariant by permutations of ; on the right-hand side, we assume that
which explains why factor disappears in the right-hand side.
The substitution above is one-to-one only if the particle does not hit twice the same obstacle. Define therefore
respectively the Markovian part and the recollision part in .
After averaging over the obstacle configuration , the contribution of the -th term in is, to leading order in :
It is dominated by
which is the general term of a converging series.
Passing to the limit as , so that , one finds (by dominated convergence in the series) that
which is the Duhamel series giving the solution of the Lorentz kinetic equation.
Hence, we have proved that
where is the solution of the Lorentz kinetic equation. One can check by a straightforward computation that the Lorentz collision integral satisfies the property
Integrating both sides of the Lorentz kinetic equation in the variables over shows that the solution of that equation satisfies
for each .
On the other hand, the billiard flow obviously leaves the uniform measure on (i.e. the particle number) invariant, so that, for each and each ,
We therefore deduce from Fatou’s lemma that
which concludes our sketch of the proof of Gallavotti’s theorem. ∎
Some remarks are in order before leaving Gallavotti’s setting for the Lorentz gas with the Poisson distribution of obstacles.
Assuming no external force field as done everywhere in the present paper is not as inocuous as it may seem. For instance, in the case of Poisson distributed holes — i.e. purely absorbing obstacles, so that particles falling into the holes disappear from the system forever — the presence of an external force may introduce memory effects in the Boltzmann-Grad limit, as observed by L. Desvillettes and V. Ricci .
Another remark is about the method of proof itself. One has obtained the Lorentz kinetic equation after having obtained an explicit formula for the solution of that equation. In other words, the equation is deduced from the solution — which is a somewhat unusual situation in mathematics. However, the same is true of Lanford’s derivation of the Boltzmann equation , as well as of the derivation of several other models in nonequilibrium statistical mechanics. For an interesting comment on this issue, see , on p. 75.
3. Santaló’s formula
for the geometric mean free path
From now on, we shall abandon the random case and concentrate our efforts on the periodic Lorentz gas.
Our first task is to define the Boltzmann-Grad scaling for periodic systems of spherical obstacles. In the Poisson case defined above, things were relatively easy: in space dimension , the Boltzmann-Grad scaling was defined by the prescription that the number of obstacles per unit volume tends to infinity while the obstacle radius tends to in such a way that
The product above has an interesting geometric meaning even without assuming a Poisson distribution for the obstacle centers, which we shall briefly discuss before going further in our analysis of the periodic Lorentz gas.
Perhaps the most important scaling parameter in all kinetic models is the mean free path. This is by no means a trivial notion, as will be seen below. As suggested by the name itself, any notion of mean free path must involve first the notion of free path length, and then some appropriate probability measure under which the free path length is averaged.
For simplicity, the only periodic distribution of obstacles considered below is the set of balls of radius centered at the vertices of a unit cubic lattice in the -dimensional Euclidian space.
Correspondingly, for each , we define the domain left free for particle motion, also called the “billiard table” as
Defining the free path length in the billiard table is easy: the free path length starting from in the direction is
Obviously, for each the free path length in the direction can be extended continuously to
where denotes the unit normal vector to at the point pointing towards .
With this definition, the mean free path is the quantity defined as
where the notation designates the average under some appropriate probability measure on .
A first ambiguity in the notion of mean free path comes from the fact that there are two fairly natural probability measures for the Lorentz gas.
The first one is the uniform probability measure on
that is invariant under the billiard flow — the notation designates the -dimensional uniform measure of the unit sphere . This measure is obviously invariant under the billiard flow
with denoting the reflection with respect to the hyperplane .
The second such probability measure is the invariant measure of the billiard map
where is the unit inward normal at , while is the -dimensional surface element on , and
The billiard map is the map
which obviously passes to the quotient modulo -translations:
In other words, given the position and the velocity of a particle immediatly after its first collision with an obstacle, the sequence is the sequence of all collision points and post-collision velocities on that particle’s trajectory.
With the material above, we can define a first, very natural notion of mean free path, by setting
Notice that, for -a.e. , the right hand side of the equality above is well-defined by the Birkhoff ergodic theorem. If the billiard map is ergodic for the measure , one has
for -a.e. .
Now, a very general formula for computing the right-hand side of the above equality was found by the great spanish mathematician L. A. Santaló in 1942. In fact, Santaló’s argument applies to situations that are considerably more general, involving for instance curved trajectories instead of straight line segments, or obstacle distributions other than periodic. The reader interested in these questions is referred to Santaló’s original article .
One finds that
where is the unit ball of and its -dimensional Lebesgue measure.
In fact, one has the following slightly more general
Lemma 3.1 (H.S. Dumas, L. Dumas, F. Golse ).
For such that , one has
Santaló’s formula is obtained by setting in the identity above, and expressing both integrals in terms of the normalized measures and .
For each one has
Hence solves the transport equation
Since and , one has
Integrating both sides of the equality above, and applying Green’s formula shows that
— beware the unusual sign in the right-hand side of the second equality above, coming from the orientation of the unit normal , which is pointing towards . ∎
With the help of Santaló’s formula, we define the Boltzmann-Grad limit for the Lorentz gas with periodic as well as random distribution of obstacles as follows:
The Boltzmann-Grad scaling for the periodic Lorentz gas in space dimension corresponds with the following choice of parameters:
|distance between neighboring lattice points|
|mean free path|
Santaló’s formula indicates that one should have
Therefore, given an initial particle density , we define to be
where is the billiard flow in with specular reflection on .
Notice that this formula defines for only, as the particle density should remain for all time in the spatial domain occupied by the obstacles. As explained in the previous section, this is a set whose measure vanishes in the Boltzmann-Grad limit, and we shall always implicitly extend the function defined above by for .
Since is a bounded function on , the family defined above is a bounded family of . By the Banach-Alaoglu theorem, this family is therefore relatively compact for the weak-* topology of .
Problem: to find an equation governing the weak-* limit points of the scaled number density as .
In the sequel, we shall describe the answer to this question in the -dimensional case (.)
4. Estimates for the distribution of free-path lengths
In the proof of Gallavotti’s theorem for the case of a Poisson distribution of obstacles in space dimension , the probability that a strip of width and length does not meet any obstacle is , where is the parameter of the Poisson distribution — i.e. the average number of obstacles per unit surface.
This accounts for the loss term
in the Duhamel series for the solution of the Lorentz kinetic equation, or of the term on the right-hand side of that equation written in the form
Things are fundamentally different in the periodic case. To begin with, there are infinite strips included in the billiard table which never meet any obstacle.
The contribution of the 1-particle density leading to the loss term in the Lorentz kinetic equation is, in the notation of the proof of Gallavotti’s theorem
The analogous term in the periodic case is
where is the free-path length in the periodic billiard table starting from in the direction .
Passing to the weak-* limit as reduces to finding
— possibly after extracting a subsequence . As we shall see below, this involves the distribution of under the probability measure introduced in the discussion of Santaló’s formula — i.e. assuming the initial position and direction to be independent and uniformly distributed on .
We define the (scaled) distribution under of free path lengths to be
Notice the scaling in this definition. In space dimension , Santaló’s formula shows that
and this suggests that the free path length is a quantity of the order of . (In fact, this argument is not entirely convincing, as we shall see below.)
In any case, with this definition of the distribution of free path lengths under , one arrives at the following estimate.
The lower bound and the upper bound in this theorem are obtained by very different means.
The lower bound, on the other hand, is obtained by working in physical space. Specifically, one uses a channel technique, introduced independently by P. Bleher  for the diffusive scaling.
This lower bound alone has an important consequence:
For each , the average of the free path length (mean free path) under the probability measure is infinite:
Indeed, since is the distribution of under , one has
Recall that the average of the free path length unded the “other” natural probability measure is precisely Santaló’s formula for the mean free path:
One might wonder why averaging the free path length under the measures and actually gives two so different results.
First observe that Santaló’s formula gives the mean free path under the probability measure concentrated on the surface of the obstacles, and is therefore irrelevant for particles that have not yet encountered an obstacle.
Besides, by using the lemma that implies Santaló’s formula with , one has
Whenever the components are independent over , the linear flow in the direction is topologically transitive and ergodic on the -torus, so that for each and . On the other hand, for some (the periodic billiard table) whenever belongs to some specific class of unit vectors whose components are rationally dependent, a class that becomes dense in as