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 $r$, placed according to a Poisson point process of density $\varrho$, in the limit $\varrho\to\infty$, $r\to0$, $\varrho r^{2}\to1$ up to time scales of order $T=o(r^{-2}{|\log r|}^{-2})$. To our knowledge this represents the first significant progress towards solving this problem in classical nonequilibrium statistical physics, since the groundbreaking work of Gallavotti (1970), Spohn (1978) 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.


Introduction
We consider the Lorentz gas with randomly placed spherical hard core scatterers in R d . That is, place spherical balls of radius r and infinite mass centred on the points of a Poisson point process of intensity in R d , where r d is sufficiently small so that with positive probability there is free passage out to infinity, and define t → X r, (t) ∈ R d 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 [19].
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 [3] where weak convergence of the process t → X r, (t) to a continuous time random walk t → Y (t) (called Markovian flight process) was established in the Boltzmann-Grad (a.k.a. low density) limit r → 0, → ∞, r d−1 → 1, in compact time intervals t ∈ [0, T ], with T < ∞, in the annealed [9,10,17,18], respectively, quenched [3] 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: r → 0, → ∞, r d−1 → 1 and T = T (r) → ∞. Thus while the diffusive limit (1) with fixed r 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.

The random Lorentz gas
We define now more formally the random Lorentz process. Place spherical balls of radius r and infinite mass centred on the points of a Poisson point process of intensity in R d , and define the trajectory t → X r, (t) ∈ R d of a particle moving among these scatterers as follows: -If the origin is covered by a scatterer then X r, (t) ≡ 0.
-If the origin is not covered by a scatterer then t → X r, (t) is the trajectory of a point-like particle starting from the origin with random velocity sampled uniformly from the unit sphere S d−1 and flying with constant speed between successive elastic collisions on any one of the fixed, infinite mass scatterers.
The randomness of the trajectory t → X r, (t) (when not identically 0) 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 t → X r, (t) is well defined. Due to elementary scaling and percolation arguments P the moving particle is not trapped in a compact domain = ϑ d ( r d ), where ϑ d : R + → [0, 1] is a percolation probability which is (i) monotone non-increasing; (ii) continuous except for one possible jump at a positive and finite critical value u c = u c (d) ∈ (0, ∞); (iii) vanishing for u ∈ (u c , ∞) and positive for u ∈ (0, u c ); (iv) lim u→0 ϑ d (u) = 1. We assume that r d < u c . In fact, in the Boltzmann-Grad limit considered in this paper (see (3) below) we will have r d → 0.
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.
(Q) 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.
(AQ) Averaged-quenched (a.k.a. annealed ) limit: Averaging over the random initial velocity of the particle and the random placements of the scatterers.

The Boltzmann-Grad limit
The Boltzmann-Grad limit is the following low (relative) density limit of the scatterer configuration: where v d−1 is the area of the (d − 1)-dimensional unit disc. In this limit the expected free path length between two successive collisions will be 1. Other choices of lim r d−1 ∈ (0, ∞) 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 EXP (1) and the jump in velocity after the free flight happens in a Markovian way with transition kernel where dv is the surface element on S d−1 and σ : S d−1 × S d−1 toR + is the normalised differential cross section of a spherical hard core scatterer, computable as Note that in 3-dimensions the transition probability (4) of velocity jumps is uniform. That is, the outgoing velocity v out is uniformly distributed on S 2 , independently of the incoming velocity v in . 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 R d , see [1]. The process t → Y (t) on the right hand side is the Markovian random flight process consisting of independent free flights of EXP (1)-distributed length, with Markovian velocity changes according to the scattering transition kernel (4). A formal construction of the process t → Y (t) is given in section 2.1. The limit (6), valid in any compact time interval t ∈ [0, T ], T < ∞, is rigorously established in the averaged-quenched setting in [9,10,17,18], and in the quenched setting in [3]. In [17] more general point processes of the scatterer positions, with sufficiently strong mixing properties are considered.
The limiting Markovian flight process t → Y (t) is a continuous time random walk. Therefore, by taking a second, diffusive limit after the Boltzmann-Grad limit (6), Donsker's theorem (see [1]) yields indeed the invariance principle, as T → ∞, where t → W (t) is the Wiener process in R d of nondegenerate variance. The variance of the limiting Wiener process W 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 d = 3 rather than general dimension.

Results
In the rest of the paper we assume = (r) = πr −2 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 t → Y (t), and the quenched-averaged (annealed) Lorentz process t → X r (t). 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 T r −1 , the Markovian flight and Lorentz exploration processes stay together. Theorem 1. Let T = T (r) be such that lim r→0 T (r) = ∞ and lim r→0 rT (r) = 0. 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 T < ∞. 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. Theorem 2. Let T = T (r) be such that lim r→0 T (r) = ∞ and lim r→0 r 2 |log r| 2 T (r) = 0. Then, for any δ > 0, and hence as r → 0, in the averaged-quenched sense. On the right hand side of (10) W is a standard Wiener process of variance 1 in R 3 .
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 T = T (r) = o(r −2 |log r| −2 ).

Remarks on dimension:
(1) Our proof is not valid in 2-dimensions for two different reasons: (a) Probabilistic estimates at the core of the proof are valid only in the transient dimensions of random walk, d ≥ 3.
(b) A subtle geometric argument which will show up in sections 6.4-6.6 below, is valid only in d ≥ 3, as well. This is unrelated to the recurrence/transience dichotomy and it is crucial in controlling the short range recollision events in the Boltzmann-Grad limit (3).
(2) The fact that in d = 3 the differential cross section of hard spherical scatterers is uniform on S 2 , c.f. (4), (5), facilitates our arguments, since, in this case, the successive velocities of the random flight process Y (t) 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 d = 3.
The proof will be based on a coupling (that is: a joint realisation on the same probability space) of the Markovian flight process t → Y (t) and the averaged-quenched realisation of the Lorentz process t → X r (t), such that the maximum distance of their positions up to time T be small order of √ T . The Lorentz process t → X r (t) 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 X r (t) traverses yet unexplored territories, it behaves just like the Markovian flight process Y (t), 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 X r (t) returns to these positions, recollisions occur. Likewise, the area swept in the past by the Lorentz exploration process X r (t) -that is: a tube of radius r 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 U (t) :=Ẏ (t) and V r (t) :=Ẋ r (t). These are almost surely piecewise constant jump processes. The coupling is realized in such a way, that (A) At the very beginning the two velocities coincide, V r (0) = U (0).
(B) Occasionally, with typical frequency of order r 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 V r (t) changes while U (t) does not.
• Scatterings of the Markovian flight process Y (t) 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 U (t) has a jump discontinuity, while the process V r (t) stays unchanged. We will call these events shadowed scatterings of the Markovian flight process.
(C) However, shortly after the mismatch events described in item (B) 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 EXP (1)-distributed time after the mismatches. Figure  1: The above image shows a recollision (left) and a shadowing event (right). Note that after each event U and V r are no longer coupled. However at the next scattering, if possible, the velocities are recoupled.
Summarizing: The coupled velocity processes t → (U (t), V r (t)) 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 r −1 . Other, more complicated mismatches of the two processes occur only at time scales of order r −2 |log r| −2 . If all these are controlled (this will be the content of the proof) then the following hold: Up to T = T (r) = o(r −1 ), with high probability there is no mismatch whatsoever between U (t) and V r (t). That is, In particular, the invariance principle (10) also follows, with T = T (r) = o(r −1 ), rather than T = T (r) = o(r −2 |log r| −2 ). 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.
Going up to T = T (r) = o(r −2 |log r| −2 ) needs more argument. The ideas exposed in the outline (A), (B), (C) above lead to the following chain of bounds: In the step we use the arguments (B) and (C). Finally, choosing in the end T = T (r) = o(r −2 ) we obtain a tightly close coupling of the diffusively scaled processes t → X r (T t)/ √ T and t → Y (T t)/ √ T , (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.

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 [13] 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 [19].

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, [4]. (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 [6], under a still-not-proved technical assumption on singularities of the corresponding billiard flow.
In the case of infinite horizon (e.g. the plain Z d 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 [2] indicated that in the twodimensional case super-diffusive scaling of order √ t log t is expected. A central limit theorem with this anomalous scaling was proved with full rigour in [20], for the Lorentz-particle displacement in the 2-dimensional periodic case with infinite horizon. The periodic infinite horizon case in dimensions d ≥ 3 remains open.

Boltzmann-Grad limit of the periodic Lorentz gas
The Boltzmann-Grad limit in the periodic case means spherical scatterers of radii r 1 placed on the points of the hypercubic lattice r (d−1)/d Z d . 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 [5] and [15] it is proved that in the Boltzmann-Grad limit the trajectory of the Lorentz particle in any compact time interval t ∈ [0, T ] with T < ∞ 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 [16].) As a second limit, an invariance principle is proved in [16] for this non-Markovian random flight process, with superdiffusive scaling √ t log t. 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 [5] are valid in d = 2 while those of [15] and [16] in arbitrary dimension.
Interpolating between the plain scaling limit in the infinite horizon case (open in d ≥ 3) and the kinetic limit, by simultaneously taking the Boltzmann-Grad limit and scaling the trajectory by √ T log T , where T = T (r) → ∞ with some rate, would be the problem analogous to our Theorem 1 or Theorem 2. This is widely open.

Miscellaneous
The quantum analogue of the problem of the Boltzmann-Grad limit for the random Lorentz gas was considered in [8], 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 Z d with extra scatterers placed randomly and independently within the cells of periodicity, [12]. 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.

Structure of the paper
The rest of the paper is devoted to the rigorous statement and proof of the arguments exposed in (A), (B), (C) above. Its overall structure is as follows: -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, Z, which will be simpler to work with than X.
-Section 3: We prove Theorem 1. We go through the proof of this result as it is both informative for the dynamics, and the proof of Theorem 2 in its full strength will follow partially similar lines, however with substantial differences.
Sections 4-7 are fully devoted to the proof of Theorem 2, as follows: -Section 4: We break up the process Z into independent legs. From here we state two propositions which are central to the proof. They state that (i) with high probability the process X does not differ from Z in each leg; (ii) with high probability, the different legs of the process Z 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 X and Z within a single leg. This section is longer than the others, due to the subtle geometric arguments and estimates needed in this proof.
-Section 7: We finish off the proof of Theorem 2.

Ingredients and the Markovian flight process
Let ξ j ∈ R + and u j ∈ R 3 , j = −2, −1, 0, 1, 2, . . . , be completely independent random variables (defined on an unspecified probability space (Ω, F , P)) with distributions: and let For later use we also introduce the sequence of indicators and the corresponding conditional exponential distributions EXP (1|1) respectively, We will also use the notation := ( j ) j≥0 and call the sequence the signature of the i.i.d.
The variables ξ j and u j will be, respectively, the consecutive flight length/flight times and flight velocities of the Markovian flight process t → Y (t) ∈ R 3 defined below.
Denote, for n ∈ Z + , t ∈ R + , That is: τ n denotes the consecutive scattering times of the flight process, ν t is the number of scattering events of the flight process Y occurring in the time interval (0, t], and {t} is the length of the last free flight before time t. Finally let We shall refer to the process t → Y (t) 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 (F t ) t≥0 of the flight process: Note that the processes n → Y n , t → Y (t) and their respective natural filtrations (F n ) n≥0 , (F t ) t≥0 , do not depend on the parameter r.
We also define, for later use, the virtual scatterers of the flight process t → Y (t). For n ≥ 0, let Here and throughout the paper we use the notation The points Y n ∈ R 3 are the centres of virtual spherical scatterers of radius r which would have caused the nth scattering event of the flight process. They do not have any influence on the further trajectory of the flight process Y , but will play role in the forthcoming couplings.

The Lorentz exploration process
Let r > 0, and = (r) = πr −2 . We define the Lorentz exploration process t → X(t) = X r (t) ∈ R 3 , coupled with the flight process t → Y (t), adapted to the filtration (F t ) t≥0 . The process t → X(t) and all upcoming random variables related to it do depend on the choice of the parameter r (and ), but from now on we will suppress explicit notation of dependence upon these parameters. The construction goes inductively, on the successive time intervals [τ n−1 , τ n ), n = 1, 2, . . . .

Start with [Step 1] and then iterate indefinitely [Step 2] and [
Step 3] below. [ Step 1] Start with Note that the trajectory of the exploration process X begins with a collision at time t = 0. This is not exactly as described previously but is of no consequence and aids the later exposition.

Go to [
Step 2]. [ Step 2] This step starts with given is a fictitious point at infinity, with inf x∈R 3 |x − | = ∞, introduced for bookkeeping reasons; The trajectory t → X(t), t ∈ [τ n−1 , τ n ), is defined as free motion with elastic collisions on fixed spherical scatterers of radius r centred at the points in S X n−1 . At the end of this time interval the position and velocity of the Lorentz exploration process are X(τ n ) =: X n , respectively, V (τ − n ). Go to [Step 3].

[Step 3] Let
Note that d n ≤ r.
• If d n < r then let X n := , and The process t → X(t) is indeed adapted to the filtration (F t ) 0≤t<∞ and indeed has the averagedquenched distribution of the Lorentz process.
Our notation is fully consistent with the one used for the markovian process Y : X n := X(τ n ) and

Mechanical consistency and compatibility of piece-wise linear trajectories in R 3
The key notion in the exploration construction of section 2.2 was mechanical r-consistency, and r-compatibility of finite segments of piece-wise linear trajectories in R 3 , which we are going to formalize now, for later reference. Let be given and define for j = 0, . . . , n, and for t ∈ [τ j , τ j+1 ], j = 0, . . . , n, We call the piece-wise linear trajectory respectively, min Note, that by formal definition the minimum distance on the left hand side can not be strictly larger than r. Given two finite pieces of mechanically r-consistent trajectories Z a (t) : τ − a,0 < t < τ + a,na and respectively. It is obvious that given a mechanically r-consistent trajectory, any non-overlapping parts of it are pairwise mechanically r-compatible, and given a finite number of non-overlapping mechanically r-consistent pieces of trajectories which are also pair-wise mechanically r-compatible their concatenation (in the most natural way) is mechanically r-consistent.

An auxiliary process
It will be convenient to introduce a third, auxiliary process t → Z(t) ∈ R 3 , and consider the joint realization of all three processes t → (Y (t), X(t), Z(t)) 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 t → Z(t) will be a forgetful version of the true physical process t → X(t) 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 T = T (r) = o(r −2 |log r| −2 ) the trajectories of the forgetful process Z(t) and the true physical process X(t) coincide, and (b) the forgetful process Z(t) and the Markovian process Y (t) stay sufficiently close together with probability tending to 1 (as r → 0). Thus, the invariance principle (7) can be transferred to the true physical process X(t), thus yielding the invariance principle (10).
Define the following indicator variables: Before constructing the auxiliary process t → Z(t) we prove the following Lemma 1. There exists a constant C < ∞ such that for any sequence of signatures = ( j ) j≥1 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 Then, clearly, It is straightforward that the indicators η j : 1 ≤ j < ∞ , and likewise, the indicators η j : 1 ≤ j < ∞ , 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 r → 0, with probability tending to 1, up to time of order T = T (r) = o(r −2 |log r| −1 ) it will not occur that two neighbouring or next-neighbouring η-s happen to take the value 1 which would obscure the following construction.
The process t → Z(t) is constructed on the successive intervals [τ j−1 , τ j ), j = 1, 2, . . . , as follows: • (Direct recollision with the last seen scatterer.) If η j = 0 and η j = 1 then, in the time interval τ j−1 ≤ t ≤ τ j the trajectory t → Z(t) is defined as that of a mechanical particle starting with initial position Z(τ j−1 ), initial velocityŻ(τ + j−1 ) = u j and colliding elastically with two infinite-mass spherical scatterers of radius r centred at the points Consistently with the notations adopted for the processes Y (t) and X(t), we denote  (1), 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.

Interferences
Let t → Y (t) and t → Y * (t) be two independent Markovian flight processes. Think about Y (t) as running forward and Y * (t) as running backwards in time. (Note, that the Markovian flight process has invariant law under time reversal.) Define the following events In words W j is the event that the virtual collision at Y j is shadowed by the past path. While W j is the event that in the time interval (τ j−1 , τ j ) 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.

Occupation measures (Green's functions)
Define the following occupation measures (Green's functions): Obviously,

Bounds
Lemma 2. The following identities and upper bounds hold: where with appropriately chosen C < ∞ and c > 0.
Proof of Lemma 2. The identity h = g is a direct consequence of the flight length ξ being EXP (1)-distributed. The distribution g has the explicit expression from which the the upper bound (24) follows.
(25) then follows from (23) and standard Green's function estimate for a random walk with step distribution g.
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 ∈ {0, 1}, respectively ∈ {0, 1} N : and state the conditional version of Lemma 2: Lemma 3. The following upper bounds hold uniformly in ∈ {0, 1} N : with K(x) and L(x) as in (26), with appropriately chosen constants C < ∞ and c > 0.
Proof of Lemma 3. Noting that the proof of Lemma 3 follows very much the same lines as the proof of Lemma 2. We omit the details.

Moreover, straightforward computations yield
Proof of Lemma 4. The bounds (29) readily follow from explicit computations. We omit the details.
We conclude this section with the following consequence of the above arguments and computations.
Corollary 1. There exists a constant C < ∞ such that for any j ≥ 1: 3.5 No mismatching -up to T ∼ o(r −1 ) Define the stopping time and note that by construction Lemma 5. Let T = T (r) be such that lim r→0 T (r) = ∞ and lim r→0 rT (r) = 0. Then Proof of Lemma 5.
where C < ∞ and c > 0. The first term in the middle expression of (33) is bounded by union bound and (30) of Corollary 1. In bounding the second term we use a large deviation upper bound for the sum of independent EXP (1)-distributed ξ j -s. Finally, (32) readily follows from (33).

Beyond the naïve coupling
The forthcoming parts of the paper rely on the joint realization (coupling) of the three processes t → Y (t), X(t), Z(t) as described in section 2. In particular, recall the construction of the process t → Z(t) from section 2.4.
A key observation is that due to the rules of construction of the process t → Z(t) exposed in section 2.4, the legs (θ n ; Z n (t) : 0 ≤ t ≤ θ n ) , n ≥ 0, of the auxiliary process t → Z(t) are also independently constructed from the packs (35), following the rules in section 2.4. Note, that the restrictions |y j−1 | < 1 in (18) were imposed exactly in order to ensure this independence of the legs (36). Therefore we will construct now the auxiliary process t → Z(t) and its time reversal t → Z * (t) from an infinite sequence of independent packs (35). In order to reduce unnecessary complications of notation from now on we assume min{ξ 0 , ξ 1 } > 1.
Remark: In order to break up the auxiliary process t → Z(t) into independent legs the choice of simpler stopping times Γ n := min{j ≥ Γ n−1 + 1 : min{ξ j , ξ j+1 } > 1}, would work. However, we need the slightly more complicated stoppings Γ n , given in (34), for some other reasons which will become clear towards the end of section 4.2 and in the statement and proof of Lemma 6.
and y j as in (13). Let Note that γ can not assume the values {1, 3, 4}. Call a pack, and keep the notation τ j := j k=1 ξ k , and θ := τ γ . The forward leg is constructed from the pack according to the rules given in section 2.4. We will also denote These are the discrete steps, respectively, the terminal position of the leg. It is easy to see that the distributions of γ and θ are exponentially tight: there exist constants C < ∞ and c > 0 such that for any s ∈ [0, ∞) The backwards leg is constructed from the pack as where the backwards pack * := (γ; (ξ γ−j , −u γ−j ) : 0 ≤ j ≤ γ) is the time reversion of the pack . Note that the forward and backward packs, and * , are identically distributed but the forward and backward processes t → Z(t) : 0 ≤ t ≤ θ and t → Z * (t) : 0 ≤ t ≤ θ are not. The backwards process t → Z * (t) could also be defined in stepwise terms, similar (but not identical) to those in section 2.4, but we will not rely on these step-wise rules and therefore omit their explicit formulation. Consistent with the previous notation, we denote Note, that due to the construction rules of the forward and backward legs, their beginning, middle and ending parts are independent, and likewise for the backwards process Z * , This fact will be of crucial importance in the proof of Proposition 2, section 5.2 below. This is the reason (alluded to in the remark at the end of section 4.1) we chose the somewhat complicated stopping time as defined in (37).
In order to construct the concatenated forward and backward processes t → Z(t), t → Z * (t), 0 ≤ t < ∞, we first define for n ∈ Z + , respectively t ∈ R + Γ n := Note that Ξ n and Ξ * n are random walks with independent steps; t → Z(t), 0 ≤ t < ∞, is exactly the Z-process constructed in section 2.4, with Z n = Z(τ n ), 0 ≤ n < ∞. Similarly, t → Z * (t), 0 ≤ t < ∞, is the time reversal of the Z-process and Z * n = Z * (τ n ), 0 ≤ n < ∞. Theorem 2 will follow from Propositions 1 and 2 of the next two sections.

Inter-leg mismatches
Let t → Z(t) be a forward Z-process built up as concatenation of legs, as exposed in section 4.3 and define the following events In words W j is the event that a collision occuring in the j-th leg is shadowed by the past path. While W j is the event that within the j-th leg the Z-trajectory bumps into a scatterer placed in an earlier leg. That is, W j ∪ W j is precisely the event that the concatenated first j − 1 legs and the j-th leg are mechanically r-incompatible (see section 2.3).
The following proposition indicates that on our time scales there are no "inter-leg mismatches": The proof of Proposition 2 is the content of Section 5

Proof of Proposition 2
This section is purely probabilistic and of similar spirit as section 3. The notation used is also similar. However, similar is not identical. The various Green's functions used here, although denoted g, h, G, H, as in section 3, are similar in their rôle but not the same. The estimates on them are also different.

Occupation measures (Green's functions)
Let now t → Z * (t), 0 ≤ t < ∞, be a backward Z * -process and t → Z(t), 0 ≤ t ≤ θ, a forward one-leg Z-process, assumed independent. In analogy with the events W j and W j defined in (44) we define It is obvious that On the other hand, by the union bound and independence we have Therefore, in view of (46) we have to control the mean occupation time measures appearing on the right hand side of (47).

Define the following mean occupation measures (Green's functions): for
It is obvious that (48)

Bounds
Lemma 6. The following upper bounds hold: Proof of Lemma 6. The proof of the bounds (49) hinges on the decompositions (40) and (41) of the forward and backward legs into independent parts. Let and Due to the exponential tail of the distribution of γ and θ, (39), there are constants C < ∞ and c > 0 such that for any s < ∞ and furthermore, From the independent decompositions (41) and (40) it follows that The bounds (49) readily follow from the explicit expressions (52), the convolutions (55) and the bounds (53) and (54). The bound (50) is a straightforward Green's function bound for the the random walk Ξ * n defined in (42), by noting that the distribution of the i.i.d. steps Z * k of this random walk has bounded density and exponential tail decay.
Remark: On the difference between Lemmas 2 and 6. Note the difference between the upper bounds for g in (24), respectively, (49), and on G in (25), respectively, (51). These are important and are due to the fact that the length first step in a Zor Z * -leg is distributed as (ξ | ξ > 1) ∼ EXP (1|0) rather than ξ ∼ EXP (1).

Computation
According to (47) Lemma 7. In dimension d = 3 the following bounds hold, with some C < ∞ Proof of Lemma 7. The bounds (57) (similarly to the bounds (29)) readily follow from explicit computations which we omit.
Proof of Proposition 2. Proposition 2 now follows by inserting the bounds (57) and one of the bounds in (29) into equations (56).
Note that by construction η 1 = η 2 = η 3 = η γ = 0, so the sums on the left hand side go actually from 4 to γ − 1 . We stated and prove these bounds in their increasing order of complexity: (58) (proved in section 6.1) and (59) (proved in section 6.2) are of purely probabilistic nature while (60) (proved in sections 6.3-6.7) also relies on the the finer geometric understanding of the mismatch events η j = 1 and η j = 1.

Proof of (59)
First note that by construction of the processes (X (t), Z(t)) : 0 − < t < θ + the following identities hold: And, hence By simple geometric inspection we see And therefore, max P min On the other hand, from the conditional Green's function computations of section 3, in particular from Lemma 3, we get max P min Putting (61), (62) and (63) together yields and hence, taking expectation over , we get (59).
Note that on the event {η j = δ j,k : 1 ≤ j ≤ γ} we have Z (k) (t) ≡ Z(t), 0 − < t < θ + . We will show that and hence Then, taking expectation over we get (60). In order to prove (64) first write and note that the three parts are independent -even if the events { η k = 1}, respectively, { η k = 1} ∩ { η k = 0} are specified. From the construction of the processes (X (t), Z (k) (t)) : a,a , 1 ≤ a ≤ 3, the event that the a-th part of the decomposition (65) is  mechanically r-inconsistent, and by A a,b = A b,a , 1 ≤ a, b ≤ 3, a = b, the event that the a-th and b-th parts of the decomposition (65) are mechanically r-incompatible -in the sense of the definitions (16) and (17) in section 2.3. In order to prove (64) we will have to prove appropriate upper bounds on the conditional probabilities These are altogether 12 bounds. However, some of them are formally very similar. A 3,3 and A (k) 1,3 do not involve the middle part and therefore do not rely on the geometric arguments of the forthcoming sections 6.4-6.6. Applying directly (19), (27), (29) and similar procedures as in section 3.4, without any new effort we get We omit the repetition of these details. The remaining six bounds rely on the geometric arguments of sections 6.4-6.6 and, therefore, are postponed to section 6.7
is the trajectory of a mechanical particle, with initial position Z r (ξ) and initial velocity˙ Z r (ξ + ) = v, bouncing elastically between two infinite-mass spherical scatterers centred at r e−u |e−u| , respectively, ξu + r u−v |u−v| , and, eventually, flying indefinitely with constant terminal velocity.
The trapping time β r , β r ∈ R + and escape (terminal) velocity w r , w r ∈ S 2 of the process Z r (t), respectively, Z r (t), are β r := 0, w r := u, β r := sup{s < ∞ :˙ Z r (ξ + s + ) =˙ Z r (ξ + s − )}, w r :=˙ Z r (ξ + β + r ). (69) Note that β r ≥ σ r . The relation of the middle segment of (65) to Z r and Z r is the following: where ∼ stands for equality in distribution. So, in order to prove (64) we have to prove some subtle estimates for the processes Z r amd Z r . The main estimates are collected in Proposition 3 below Proposition 3. There exists a constant C < ∞, such that for all r < 1 and s ∈ (0, ∞), the following bounds hold: Remarks: The bound (71) is sharp in the sense that a lower bound of the same order can be proved. In contrast, we think that the upper bound in (72) is not quite sharp. However, it is sufficient for our purposes so we don't strive for a better estimate.
The following consequence of Proposition 3 will be used to prove (60).

Corollary 2.
There exists a constant C < ∞ such that the following bounds hold: ≤ Cr max{s |log s| 2 , r |log r| 2 } Proposition 3 and its Corollary 2 are proved in sections 6.5, respectively, 6.6.
6.5 Geometric estimates ctd: Proof of Proposition 3

Preparations
Beside the probability measure µ (see (68)) we will also need the flat Lebesgue measure on D, For r > 0 we define the dilation map D r : D → D as and note that In the forthcoming steps all events in A r and A r will be mapped by the inverse dilation D −1 r = D r −1 into A 1 , respectively, A 1 . Therefore, in order to simplify notation we will use A := A 1 and The dilation D r transforms the measures µ as follows. Given an event E ⊂ D, and hence, for any event E ⊂ D and anyh < ∞ The following simple observation is of paramount importance in the forthcoming arguments: Proposition 4. In dimension 3 (and more) Proof of Proposition 4. Obviously, Since, in dimension 3, the claim follows by integrating over h ∈ R + .
Remark: In 2-dimension, the corresponding sets A, A have infinite Lebesgue measure and, therefore, a similar proof would fail. Due to (80) in 3-dimensions the following conditional probability measures make sense and, moreover, due to (79) and (80), for any event In a technical sense, we will only use the upper bound in (79), and (80). In view of the upper bound in (79), in order to prove (71), (72) and (73) we need, in turn, Here, and in the rest of this section, we use the simplified notation w := w 1 , w := w 1 , β := β 1 .

Proof of (81)
Proof. This is straightforward. Recall (69): w(u, h, v) = u. For easing notation let and note that for any t ∈ R + {u ∈ S 2 : 0 ≤ ϑ ≤ t} ≤ C min{t 2 , 1}, with some explicit C < ∞. Then,  Figure 3: Above we show a 3 dimensional example of the geometric labelling used in this section. The Z trajectory enters with velocity e from beneath the relevant plane (the dotted line represents motion below the plane). After which the particle remains above the plane.
Let a and b be the vectors in R 3 pointing from the origin to the centre of the spherical scatterers of radius 1, on which the first, respectively, the second collision occurs: and n the unit vector orthogonal to the plane determined by a and b, pointing so, that e · n > 0: are independent and distributed as Therefore, The last step follows from explicit computations which we omit. Finally, (87), (88) and (89) yield (82).
Proof of (83). We proceed with the first (sharper) bound in (86) (the second (weaker) bound would yield only upper bound of order s −1/2 on the right hand side of (82)): Bounding the first term on the right hand side of (90) is straightforward: Concerning the second term on the right hand side of (90), this has exactly been done in the proof of (82) above, ending in (89) -with the rôle of s and s −1 swapped. (90), (91) and (89) yield (73).

Geometric estimates ctd: Proof of Corollary 2
We start with the following straightforward geometric fact.
Lemma 8. Let e, w ∈ S 2 and x ∈ R 3 . Then Proof of Lemma 8. This is elementary 3-dimensional geometry. We omit the details.
The bounds in (74) and (75) follow from applying (92) and (71), bearing in mind that the distribution density of ξ k−2 and ξ k is bounded. Since these are very similar we will only prove (74) here.
In the first step we used (93). The second step follows from the representation (70). The third step relies on (92) and on uniform boundedness of the distribution density of ξ − (which is either EXP (1|1) or EXP (1|0), depending on the value of k−2 ). Finally, the last calculation is based on (71).
= min min Here, and in the rest of this proof, β and w denote the trapping time and escape direction of the recollision sequence: To bound the first expression on the right hand side of (94) we first observe that by the triangle inequality Applying the representation and bounds developed in sections 6.4, 6.5, ≤ Cr 2 + Crs + Cr 2 |log r| .
In the first step we used (95). The second step follows from the representation (70). The third step relies on on uniform boundedness of the distribution density of ξ − (which is either EXP (1|1) or EXP (1|0), depending on the value of k−2 ). Finally, the last step follows from explicit calculation, using (79).
Proof of (77). We proceed very similarly as in the proof of (76).
≥ min min To bound the first expression on the right hand side of (99) we first observe that by the triangle inequality Using in turn (100), (70), (73) and explicit computation based on uniform boundedness of the distribution density of ξ + (which is either EXP (1|1) or EXP (1|0), depending on the value of k ) we write ≤ Cr 2 + Crs + Cr 2 |log r| 2 .
The second term on the right hand side of (99) is bounded in a very similar way as the analogous second term on the right hand side of (94), see (97)-(98). Without repeating these details we state that Eventually, from (99), (101) and (102) we obtain (77).

Proof of (60) -concluded
Recall the events A It remains to prove Since the cases b = 1 and b = 3 are formally identical we will go through the steps of proof with b = 3 only. In order to do this we first define the necessary occupation time measures (Green's functions). For A ⊂ R 3 , define the following occupation time measures for the last part of (65) Similarly, define the following occupation time measures for the middle part of (65) Using the independence of the middle and last parts in the decomposition (65), similarly as (22) or (47), following bounds are obtained Due to (28) of Lemma 3 by direct computations the following upper bounds hold where C < ∞ is an appropriately chosen constant and F : R + → R, F (u) := r1{0 ≤ u < r} + r 3 u 2 1{r ≤ u < 1} + Finally, we also have the global bounds We will prove the upper bound (104) for the first term on the right hand side of the first line in (105). The other four terms are done in very similar way. First we split the integral as and note that due to (106) and (108) the second term on the right hand side is bounded as To bound the first term on the right hand side of (109) we proceed as follows In the first step we have used (106). The second step is an integration by parts. In the third step we use (107), (108) and the explicit form of the function F . The last step is explicit integration. Finally, (109), (110), (111) and identical comoputations for the second term on the right hand side of the first line in (105) yield the first inequality in (104). The second line of (104) for b = 3 is proved in an identical way, which we omit to repeat. The cases b = 1 is done in a formally identical way.
Denote θ n , ((Y n (t), Z n (t)) : 0 ≤ t ≤ θ n ) the pair of Y and (forward) Z-processes constructed from them and Beside these two we now define yet another auxiliary process t → X (t) as follows: (X n (t) : 0 ≤ t ≤ θ n ) is the Lorentz exploration process constructed with data from (Y n (t) : 0 ≤ t ≤ θ n ) and incoming velocity u n,0 = u 0 if n = 1, X n−1 (θ − n−1 ) if n > 1.
Remark: Actually, (113) holds under the much weaker condition lim r→∞ r log log T = 0. This can be achieved by applying the LIL rather than a WLLN type of argument to bound max 0≤t≤T |Y (t) − Z(t)| in the proof of Lemma 10, below. However, since the condition of Lemma 9 can not be much relaxed, in the end we would not gain much with the extra effort.
Proof of Lemma 9.
where C < ∞ and c > 0. The first term on the right hand side of (114) is bounded by union bound and (43) from Proposition 1. Likewise, the second term is bounded by union bound and (45) of Propositions 2. In bounding the third term we use a large deviation upper bound for the sum of independent θ j -s. Finally, (112) readily follows from (114).
Proof of Lemma 10. Note first that with ν T and η j defined in (15), respectively, (18). Hence, with C < ∞ and c > 0. The first term on the right hand side of (115) is bounded by Markov's inequality and the straightforward bound E η j ξ j ≤ Cr.
The bound on the second term follows from a straightforward large deviation estimate on ν T ∼ P OI(T ). Finally, (113) readily follows from (115).
(9) is direct consequence of Lemmas 9 and 10 and this concludes the proof of Theorem 2.