Invariance Principle for the Random Lorentz Gas – Beyond the Boltzmann-Grad Limit
We prove an invariance principle for a random Lorentz-gas particle in 3 dimensions under the Boltzmann-Grad limit and simultaneous diffusive scaling. That is, for the trajectory of a point-like particle moving among infinite-mass, hard-core, spherical scatterers of radius , placed according to a Poisson point process of density , in the limit , , up to time scales of order . To our knowledge this represents the first significant progress towards solving this problem in classical nonequilibrium statistical physics, since the groundbreaking work of Gallavotti (1969) [9, 10], Spohn (1978) [17, 18] and Boldrighini-Bunimovich-Sinai (1983) . The novelty is that the diffusive scaling of particle trajectory and the kinetic (Boltzmann-Grad) limit are taken simulataneously. The main ingredients are a coupling of the mechanical trajectory with the Markovian random flight process, and probabilistic and geometric controls on the efficiency of this coupling.
MSC2010: 60F17; 60K35; 60K37; 60K40; 82C22; 82C31; 82C40; 82C41
Key words and phrases: Lorentz-gas; invariance principle; scaling limit; coupling; exploration process.
We consider the Lorentz gas with randomly placed spherical hard core scatterers in . That is, place spherical balls of radius and infinite mass centred on the points of a Poisson point process of intensity in , where is sufficiently small so that with positive probability there is free passage out to infinity, and define to be the trajectory of a point particle starting with randomly oriented unit velocity, performing free flight in the complement of the scatterers and scattering elastically on them. A major problem in mathematical statistical physics is to understand the diffusive scaling limit of the particle trajectory
Indeed, the Holy Grail of this field of research would be to prove an invariance principle (i.e. weak convergence to a Wiener process with nondegenerate variance) for the sequence of processes in (1) in either the quenched or annealed setting (discussed in section 1.1). For extensive discussion and historical background see the surveys [18, 7, 14] and the monograph .
The same problem in the periodic setting, when the scatterers are placed in a periodic array and randomness comes only with the initial conditions of the moving particle, is much better understood, due to the fact that in the periodic case the problem is reformulated as diffusive limit of particular additive functionals of billiards in compact domains and thus heavy artillery of hyperbolic dynamical systems theory is efficiently applicable. In order to put our results in context, we will summarize very succinctly the existing results, in section 1.4.
There has been, however, no progress in the study of the random Lorentz gas informally described above, since the ground-breaking work of Gallavotti [9, 10], Spohn [17, 18] and Boldrighini-Bunimovich-Sinai  where weak convergence of the process to a continuous time random walk (called Markovian flight process) was established in the Boltzmann-Grad (a.k.a. low density) limit , , , in compact time intervals , with , in the annealed [9, 10, 17, 18], respectively, quenched  setting.
Our main result (see Theorem 2 in subsection 1.3) proves an invariance principle in the annealed setting if we take the Boltzmann-Grad and diffusive limits simultaneously: , , and . Thus while the diffusive limit (1) with fixed and remains open, this is the first result proving convergence for infinite times in the setting of randomly placed scatterers, and hence it is a significant step towards the full resolution of the problem in the annealed setting.
1.1 The random Lorentz gas
We define now more formally the random Lorentz process. Place spherical balls of radius and infinite mass centred on the points of a Poisson point process of intensity in , and define the trajectory of a particle moving among these scatterers as follows:
If the origin is covered by a scatterer then .
If the origin is not covered by a scatterer then is the trajectory of a point-like particle starting from the origin with random velocity sampled uniformly from the unit sphere and flying with constant speed between successive elastic collisions on any one of the fixed, infinite mass scatterers.
The randomness of the trajectory (when not identically ) is due to two sources: the random placement of the scatterers and the random choice of initial velocity of the moving particle. Otherwise, the dynamics of the moving particle is fully deterministic, governed by classical Newtonian laws. With probability 1 (with respect to both sources of randomness) the trajectory is well defined.
Due to elementary scaling and percolation arguments
where is a percolation probability which is (i) monotone non-increasing; (ii) continuous except for one possible jump at a positive and finite critical value ; (iii) vanishing for and positive for ; (iv) . We assume that . In fact, in the Boltzmann-Grad limit considered in this paper (see (3) below) we will have .
As discussed above, the Holy Grail of this field is a mathematically rigorous proof of invariance principle of the processes (1) in either one of the following two settings.
Quenched limit: For almost all (i.e. typical) realizations of the underlying Poisson point process, with averaging over the random initial velocity of the particle. In this case, it is expected that the variance of the limiting Wiener process is deterministic, not depending on the realization of the underlying Poisson point process.
Averaged-quenched (a.k.a. annealed) limit: Averaging over the random initial velocity of the particle and the random placements of the scatterers.
1.2 The Boltzmann-Grad limit
The Boltzmann-Grad limit is the following low (relative) density limit of the scatterer configuration:
where is the area of the -dimensional unit disc. In this limit the expected free path length between two successive collisions will be 1. Other choices of are equally legitimate and would change the limit only by a time (or space) scaling factor.
It is not difficult to see that in the averaged-quenched setting and under the Boltzmann-Grad limit (3) the distribution of the first free flight length starting at any deterministic time, converges to an and the jump in velocity after the free flight happens in a Markovian way with transition kernel
where is the surface element on and is the normalised differential cross section of a spherical hard core scatterer, computable as
Note that in -dimensions the transition probability (4) of velocity jumps is uniform. That is, the outgoing velocity is uniformly distributed on , independently of the incoming velocity .
It is intuitively compelling but far from easy to prove that under the Boltzmann-Grad limit (3)
where the symbol stands for weak convergence (of probability measures) on the space of continuous trajectories in , see . The process on the right hand side is the Markovian random flight process consisting of independent free flights of -distributed length, with Markovian velocity changes according to the scattering transition kernel (4). A formal construction of the process is given in section 2.1. The limit (6), valid in any compact time interval , , is rigorously established in the averaged-quenched setting in [9, 10, 17, 18], and in the quenched setting in . In  more general point processes of the scatterer positions, with sufficiently strong mixing properties are considered.
The limiting Markovian flight process is a continuous time random walk. Therefore, by taking a second, diffusive limit after the Boltzmann-Grad limit (6), Donsker’s theorem (see ) yields indeed the invariance principle,
as , where is the Wiener process in of nondegenerate variance. The variance of the limiting Wiener process can be explicitly computed but its concrete value has no importance.
The natural question arises whether one could somehow interpolate between the double limit of taking first the Boltzmann-Grad limit (6) and then the diffusive limit (7) and the plain diffusive limit for the Lorentz process, (1). Our main result, Theorem 2 formulated in section 1.3 gives a positive partial answer in dimension 3. Since our results are proved in three-dimensions from now on we formulate all statements in rather than general dimension.
In the rest of the paper we assume and drop the superscript from the notation of the Lorentz process.
Our results (Theorems 1 and 2 formulated below) refer to a coupling – joint realisation on the same probability space – of the Markovian random flight process , and the quenched-averaged (annealed) Lorentz process . The coupling is informally described later in this section and constructed with full formal rigour in section 2.2.
The first theorem states that in our coupling, up to to time , the Markovian flight and Lorentz exploration processes stay together.
Let be such that and . Then
Although, this result is subsumed by our main result, it shows the strength of the coupling method employed in this paper. In particular, with some elementary arguments it provides a much stronger result than Gallavotti and Spohn [9, 10, 17] which states the weak limit (6) (which follows from (8)) for any fixed . On the other hand the proof of this "naïve" result sheds some light on the structure of proof of the more sophisticated Theorem 2, which is our main result.
Let be such that and . Then, for any ,
as , in the averaged-quenched sense. On the right hand side of (10) is a standard Wiener process of variance in .
Indeed, the invariance principle (10) readily follows from the invariance principle for the Markovian flight process, (7), and the closeness of the two processes quantified in (9). So, it remains to prove (9). This will be the content of the larger part of this paper, sections 4-7.
The point of Theorem 2 is that the Boltzmann-Grad limit of scatterer configuration (3) and the diffusive scaling of the trajectory are done simultaneously, and not consecutively. The memory effects due to recollisions are controlled up to the time scale .
Remarks on dimension:
Our proof is not valid in 2-dimensions for two different reasons:
Probabilistic estimates at the core of the proof are valid only in the transient dimensions of random walk, .
The fact that in the differential cross section of hard spherical scatterers is uniform on , c.f. (4), (5), facilitates our arguments, since, in this case, the successive velocities of the random flight process form an i.i.d. sequence. However, this is not of crucial importance. The same arguments could also be carried out for other differential cross sections, at the expense of more extensive arguments. We are not going to these generalisations here. Therefore the proofs presented in this paper are valid exactly in .
The proof will be based on a coupling (that is: a joint realisation on the same probability space) of the Markovian flight process and the averaged-quenched realisation of the Lorentz process , such that the maximum distance of their positions up to time be small order of . The Lorentz process is realised as an exploration of the environment of scatterers. That is, as time goes on, more and more information is revealed about the position of the scatterers. As long as traverses yet unexplored territories, it behaves just like the Markovian flight process , discovering new, yet-unseen scatterers with rate 1 and scattering on them. However, unlike the Markovian flight process it has long memory, the discovered scatterers are placed forever and if the process returns to these positions, recollisions occur. Likewise, the area swept in the past by the Lorentz exploration process – that is: a tube of radius around its past trajectory – is recorded as a domain where new collisions can not occur. For a formal definition of the coupling see section 2.2. Let their velocity processes be and . These are almost surely piecewise constant jump processes. The coupling is realized in such a way, that
At the very beginning the two velocities coincide, .
Occasionally, with typical frequency of order mismatches of the two velocity processes occur. These mismatches are caused by two possible effects:
Recollisions of the Lorentz exploration process with a scatterer placed in the past. This causes a collision event when changes while does not.
Scatterings of the Markovian flight process in a moment when the Lorentz exploration process is in the explored tube, where it can not encounter a not-yet-seen new scatterer. In these moments the process has a jump discontinuity, while the process stays unchanged. We will call these events shadowed scatterings of the Markovian flight process.
However, shortly after the mismatch events described in item 2 above, a new jointly realised scattering event of the two processes occurs, recoupling the two velocity processes to identical values. These recouplings occur typically at an -distributed time after the mismatches.
Summarizing: The coupled velocity processes are realized in such a way that they assume the same values except for typical time intervals of length of order 1, separated by typical intervals of lengths of order . Other, more complicated mismatches of the two processes occur only at time scales of order . If all these are controlled (this will be the content of the proof) then the following hold:
Up to , with high probability there is no mismatch whatsoever between and . That is,
In particular, the invariance principle (10) also follows, with , rather than . As a by-product of this argument a new and handier proof of the theorem (6) of Gallavotti [9, 10] and Spohn [17, 18] also drops out.
In the step we use the arguments 2 and 3. Finally, choosing in the end we obtain a tightly close coupling of the diffusively scaled processes and , (9), and hence the invariance principle (10), for this longer time scale. This hand-waving argument should, however, be taken with a grain of salt: it does not show the logarithmic factor, which arises in the fine-tuning.
1.4 Summary of related work
In order to put our results in context we succinctly summarize the related most important results in the mathematically rigorous treatment of diffusion in the Lorentz gas. As Hendrik Lorentz’s seminal paper  where he proposes the periodic setting of what we call today the Lorentz gas for modelling diffusion and transport in solids was published in 1905, and the large amount of work done in this field, we can not strive for exhaustion, and mention only a (possibly subjective) selection of the mathematically rigorous results. For more comprehensive historical overview we refer the reader to the survey papers [7, 14, 18] and the monograph .
Scaling limit of the periodic Lorentz gas
As already mentioned, diffusion in the periodic setting is much better understood than in the random setting. This is due to the fact that diffusion in the periodic Lorentz gas can be reduced to study the of limit theorems of some particular additive functionals of billiard flows in compact domains. Heavy tools of hyperbolic dynamics provide the technical arsenal for the study of these problems.
The first breakthrough was the fully rigorous proof of the invariance principle (diffusive scaling limit) for the Lorentz particle trajectory in a two-dimensional periodic array of spherical scatterers with finite horizon, . (Finite horizon means that the length of the straight path segments not intersecting a scatterer is bounded from above.) This result was extended to higher dimensions in , under a still-not-proved technical assumption on singularities of the corresponding billiard flow.
In the case of infinite horizon (e.g. the plain arrangement of the spherical scatterers of diameter less than the lattice spacing) the free flight distribution of a particle flying in a uniformly sampled random direction has a heavy tail which causes a different type of long time behaviour of the particle displacement. The arguments of  indicated that in the two-dimensional case super-diffusive scaling of order is expected. A central limit theorem with this anomalous scaling was proved with full rigour in , for the Lorentz-particle displacement in the -dimensional periodic case with infinite horizon. The periodic infinite horizon case in dimensions remains open.
Boltzmann-Grad limit of the periodic Lorentz gas
The Boltzmann-Grad limit in the periodic case means spherical scatterers of radii placed on the points of the hypercubic lattice . The particle starts with random initial position and velocity sampled uniformly and collides elastically on the scatterers. For a full exposition of the long and complex history of this problem we quote the surveys [11, 14] and recall only the final, definitive results.
In  and  it is proved that in the Boltzmann-Grad limit the trajectory of the Lorentz particle in any compact time interval with fixed, converges weakly to a non-Markovian flight process which has, however, a complete description in terms of a Markov chain of the successive collision impact parameters and, conditionally on this random sequence, independent flight lengths. (For a full description in these terms see .) As a second limit, an invariance principle is proved in  for this non-Markovian random flight process, with superdiffusive scaling . Note that in this case the second limit doesn’t just drop out from Donsker’s theorem as it did in the random scatterer setting. The results of  are valid in while those of  and  in arbitrary dimension.
The quantum analogue of the problem of the Boltzmann-Grad limit for the random Lorentz gas was considered in , where the long time evolution of a quantum particle interacting with a random potential in the Boltzmann-Grad limit is studied. It is proved that the phase space density of the quantum evolution converges weakly to a the solution of the linear Boltzmann equation. This is the precise quantum analogue of the classical problem solved by Gallavotti and Spohn in [9, 10, 17, 18].
Looking into the future: Liverani investigates the periodic Lorentz gas with finite horizon with local random perturbations in the cells of periodicity: a basic periodic structure with spherical scatterers centred on with extra scatterers placed randomly and independently within the cells of periodicity, . This is an interesting mixture of the periodic and random settings which could succumb to a mixture of dynamical and probabilistic methods, so-called deterministic walks in random environment.
1.5 Structure of the paper
Section 2: We construct the Markovian flight process and the Lorentz exploration and thus lay out the coupling argument which is essential moving forward. Moreover we will introduce an auxiliary process, , which will be simpler to work with than .
Section 4: We break up the process into independent legs. From here we state two propositions which are central to the proof. They state that
(i) with high probability the process does not differ from in each leg;
(ii) with high probability, the different legs of the process do not interact (up to times of our time scales).
Section 5: We prove the proposition concerning interactions between legs.
Section 6: We prove the proposition concerning coincidence, with high probability, of the processes and within a single leg. This section is longer than the others, due to the subtle geometric arguments and estimates needed in this proof.
2.1 Ingredients and the Markovian flight process
Let and , , be completely independent random variables (defined on an unspecified probability space ) with distributions:
For later use we also introduce the sequence of indicators
and the corresponding conditional exponential distributions , respectively, , with distribution densities
We will also use the notation and call the sequence the signature of the i.i.d. -sequence .
The variables and will be, respectively, the consecutive flight length/flight times and flight velocities of the Markovian flight process defined below.
Denote, for , ,
That is: denotes the consecutive scattering times of the flight process, is the number of scattering events of the flight process occurring in the time interval , and is the length of the last free flight before time .
We shall refer to the process as the Markovian flight process. This will be our fundamental probabilistic object. All variables and processes will be defined in terms of this process, and adapted to the natural continuous time filtration of the flight process:
Note that the processes , and their respective natural filtrations , , do not depend on the parameter .
We also define, for later use, the virtual scatterers of the flight process . For , let
Here and throughout the paper we use the notation .
The points are the centres of virtual spherical scatterers of radius which would have caused the th scattering event of the flight process. They do not have any influence on the further trajectory of the flight process , but will play role in the forthcoming couplings.
2.2 The Lorentz exploration process
Let , and . We define the Lorentz exploration process , coupled with the flight process , adapted to the filtration . The process and all upcoming random variables related to it do depend on the choice of the parameter (and ), but from now on we will suppress explicit notation of dependence upon these parameters.
Note that the trajectory of the exploration process begins with a collision at time . This is not exactly as described previously but is of no consequence and aids the later exposition.
Go to 2.
This step starts with given , and , where
is a fictitious point at infinity, with , introduced for bookkeeping reasons;
for , and .
The trajectory , , is defined as free motion with elastic collisions on fixed spherical scatterers of radius centred at the points in . At the end of this time interval the position and velocity of the Lorentz exploration process are , respectively, .
Go to 3.
Note that .
If then let , and .
If then let ,and .
Go back to 2.
The process is indeed adapted to the filtration and indeed has the averaged-quenched distribution of the Lorentz process.
Our notation is fully consistent with the one used for the markovian process : and
2.3 Mechanical consistency and compatibility of piece-wise linear trajectories in
The key notion in the exploration construction of section 2.2 was mechanical -consistency, and -compatibility of finite segments of piece-wise linear trajectories in , which we are going to formalize now, for later reference.
be given and define for ,
and for , ,
We call the piece-wise linear trajectory mechanically -consistent or -inconsistent, if
Note, that by formal definition the minimum distance on the left hand side can not be strictly larger than .
Given two finite pieces of mechanically -consistent trajectories and , defined over non-overlapping time intervals: , with , we will call them mechanically -compatible or -incompatible if
It is obvious that given a mechanically -consistent trajectory, any non-overlapping parts of it are pairwise mechanically -compatible, and given a finite number of non-overlapping mechanically -consistent pieces of trajectories which are also pair-wise mechanically -compatible their concatenation (in the most natural way) is mechanically -consistent.
2.4 An auxiliary process
It will be convenient to introduce a third, auxiliary process , and consider the joint realization of all three processes on the same probability space. This construction will not be needed until section 4, but this is the optimal logical point to introduce it. The reader may safely skip to section 3 and come back here before turning to section 4.
The process will be a forgetful version of the true physical process in the sense that in its construction only memory effects by the last seen scatterers are taken into account. That is: only direct recollisions with the last seen scatterer and shadowings by the last straight flight segment are incorporated, disregarding more complex memory effects. It will be shown that
(a) up to times the trajectories of the forgetful process and the true physical process coincide, and
(b) the forgetful process and the Markovian process stay sufficiently close together with probability tending to (as ). Thus, the invariance principle (7) can be transferred to the true physical process , thus yielding the invariance principle (10).
Define the following indicator variables:
Before constructing the auxiliary process we prove the following
There exists a constant such that for any sequence of signatures the following bounds hold
Proof of Lemma 1.
Define the following auxiliary, and simpler, indicators:
Here, and in the rest of the paper we use the notation
It is straightforward that the indicators , and likewise, the indicators
, are independent among themselves and one-dependent across the two sequences. This holds even if conditioned on the sequence of signatures .
Therefore, the following simple computations prove the claim of the lemma.
We omit the elementary computational details. ∎
Lemma 1 assures that, as , with probability tending to , up to time of order it will not occur that two neighbouring or next-neighbouring -s happen to take the value which would obscure the following construction.
The process is constructed on the successive intervals , , as follows:
(No interference with the past.)
If then for , .
If , then for , .
(Direct recollision with the last seen scatterer.)
If and then, in the time interval the trajectory is defined as that of a mechanical particle starting with initial position , initial velocity and colliding elastically with two infinite-mass spherical scatterers of radius centred at the points
Consistently with the notations adopted for the processes and , we denote for .
3 No mismatches up to : Proof of Theorem 1
In this section we prove that the Markovian flight trajectory , up to time scales of order , is mechanically -consistent with probability , and therefore the coupling bound of Theorem 1 holds. On the way we establish various bounds to be used in later sections. This section is purely classical-probabilistic. It also prepares the ideas (and notation) for section 5 where a similar argument is explored in more complex form.
Let and be two independent Markovian flight processes. Think about as running forward and as running backwards in time. (Note, that the Markovian flight process has invariant law under time reversal.) Define the following events
In words is the event that the virtual collision at is shadowed by the past path. While is the event that in the time interval there is a virtual recollision with a past scatterer.
It is obvious that
On the other hand, by union bound and independence
Here and in the rest of the paper we use the notation for either cardinality or Lebesgue measure of the set , depending on context.
3.2 Occupation measures (Green’s functions)
Define the following occupation measures (Green’s functions): for
The following identities and upper bounds hold:
with appropriately chosen and .
Proof of Lemma 2.
The identity is a direct consequence of the flight length being -distributed. The distribution has the explicit expression
from which the the upper bound (24) follows.
For later use we introduce the conditional versions – conditioned on the sequence (see (14)) – of the bounds (24), (25). In this order we define the conditional versions of the Green’s functions, given , respectively :
and state the conditional version of Lemma 2:
The following upper bounds hold uniformly in :