Superdiffusion in the periodic Lorentz gas
Abstract.
We prove a superdiffusive central limit theorem for the displacement of a test particle in the periodic Lorentz gas in the limit of large times and low scatterer densities (BoltzmannGrad limit). The normalization factor is , where is measured in units of the mean collision time. This result holds in any dimension and for a general class of finiterange scattering potentials. We also establish the corresponding invariance principle, i.e., the weak convergence of the particle dynamics to Brownian motion.
2010 Mathematics Subject Classification:
37D50, 60F05, 60F17, 82C401. Introduction
The periodic Lorentz gas is one of the iconic models of “chaotic” diffusion in deterministic systems. It describes the dynamics of a testparticle in an infinite periodic array of spherically symmetric scatterers. The main results characterizing the diffusive nature of the periodic Lorentz gas have to date been mainly restricted to the twodimensional setting and hardsphere scatterers. The first seminal result on this subject was the proof of a central limit theorem for the displacement of the test particle at large times for the finitehorizon Lorentz gas by Bunimovich and Sinai [9]. For more general invariance principles see Melbourne and Nicol [23] and references therein. In the case of the infinitehorizon Lorentz gas, Bleher [6] pointed out that the meansquare displacement grows like when , as opposed to a linear growth in the finitehorizon case. The superdiffusive central limit theorem suggested in [6] was first proved by Szász and Varjú [29] for the discretetime billiard map. Dolgopyat and Chernov [15] provided an alternative proof, and established the central limit theorem and invariance principle for the billiard flow. Analogous results hold for the stadium billiard (Bálint and Gouëzel [2]) and billiards with cusps (Bálint, Chernov and Dolgopyat [1]). The difficulty in extending the above findings to dimensions greater than two lies in the possibly exponential growth of the complexity of singularities (Bálint and Tóth [3, 4], Chernov [13]) and, in the case of infinite horizon, the subtle geometry of channels (Dettmann [14], Nándori, Szász and Varjú [24]).
In the present paper we prove unconditional superdiffusive central limit theorems and invariance principles for the periodic Lorentz gas in any dimension , valid in the limit of low scatterer density (BoltzmannGrad limit) and for a general class of finiterange scattering potentials. That is, instead of fixing the radius of each scatterer and considering the long time limit as in the above cited papers, we consider here the limit and then the limit of long times, where time is measured in units of the mean collision time. It is an interesting open problem to consider the two limits , jointly.
The precise setting of our study is as follows. Let be a fixed Euclidean lattice of covolume one (such as the cubic lattice ), and define the scaled lattice . At each point in we center a sphere of radius . We consider a test particle that moves along straight lines with unit speed until it hits a sphere, where it is scattered elastically. The above scaling of scattering radius vs. lattice spacing ensures that the mean free path length (i.e., the average distance between consecutive collisions) has the limit as , where denotes the volume of the unit ball in .
In the case of the classic Lorentz gas the scattering mechanism is given by specular reflection, but as in [21] we will here also allow more general spherically symmetric scattering maps. The precise conditions will be stated in Section 2.
The position of our test particle at time is denoted by
(1.1) 
where and are position and velocity at time , and is the open unit ball in centered at the origin. We use the convention that for any boundary point we choose the outgoing velocity , i.e. the velocity after the scattering. The corresponding phase space is denoted by . For notational reasons it is convenient to extend the dynamics to by setting for all initial conditions .
We consider the time evolution of a test particle with random initial data , distributed according to a given Borel probability measure on . The following superdiffusive central limit theorem, valid for small scattering radii and large times, asserts that the normalized particle displacement at time , and measured in units of , converges weakly to a Gaussian distribution.
Theorem 1.1.
Let and fix a Euclidean lattice of covolume one. Assume is distributed according to an absolutely continuous Borel probability measure on . Then, taking first and then , we have
(1.2) 
where is a centered normal random variable in with identity covariance matrix, and
(1.3) 
Here denotes the Riemann zeta function. Recall that the weak convergence (1.2) holds if and only if
(1.4) 
for any bounded continuous .
Theorem 1.1 will follow from its descretetime analogue, Theorem 1.2. Let us denote by () the location where the test particle with initial condition leaves the th scatterer. It is natural in this setting to assume . By the translational invariance of the lattice, we may in fact assume without loss of generality . For given exit velocity , we write
(1.5) 
and stipulate in the following that the random variable is uniformly distributed in the unit disc orthogonal to . The uniform distribution is the natural invariant measure for the discrete time dynamics.
Theorem 1.2.
Let and as above. Assume is distributed according to an absolutely continuous Borel probability measure on . Then, taking first and then , we have
(1.6) 
with
(1.7) 
The above results generalise to functional central limit theorems, also known as invariance principles. Denote by the space of curves starting at the origin. We fix a metric on by defining the distance between two curves and by . The topology generated by open balls in this metric is called the uniform topology. A sequence of random curves in converges weakly to (), if for any bounded continuous we have .
The following theorem, which is the main result of this paper, states that for the same random initial data as in Theorem 1.1, the random curves
(1.8) 
converge weakly to the standard Brownian motion in with unit covariance matrix .
Theorem 1.3.
Let and fix a Euclidean lattice of covolume one. Assume is distributed according to an absolutely continuous Borel probability measure on . Then, taking first and then , we have
(1.9) 
As in the case of Theorem 1.1, we derive Theorem 1.3 as a corollary of its discretetime analogue, Theorem 1.4. By linearly interpolating between the position variables , we obtain the piecewise linear curve
(1.10) 
where denotes the fractional part of . We rescale the curve by setting
(1.11) 
We then have the following generalization of Theorem 1.2.
Theorem 1.4.
Let and a Euclidean lattice of covolume one. Assume is distributed according to an absolutely continuous Borel probability measure on . Then, taking first and then , we have
(1.12) 
The starting point of our analysis is the paper [21], which proves that, for every fixed , the limit in (1.2) (resp. (1.6)) exists and is given by a continuoustime (resp. discretetime) Markov process. The main objective of the present study is therefore to prove a superdiffusive central limit theorem, as well as an invariance principle, for each of these Markov processes. The central limit theorem is stated as Theorem 3.2 in Section 3 after a brief survey of the relevant results from [21]. The subsequent sections of the paper are devoted to the proof of Theorem 3.2. The invariance principles stated in Theorems 1.3 and 1.4 follow from the results in Sections 12–14.
2. The scattering map
We now specify the conditions on the scattering map that are assumed in Theorems 1.1–1.4. These are the same as in [21], with the additional simplifying assumption that the scattering map preserves angular momentum, cf. [21, Remark 2.3]. We describe the scattering map in units of , i.e., the scatterer is represented as the open unit ball . Set
(2.1) 
and consider the scattering map
(2.2) 
The incoming data is denoted by , where is the velocity of the particle before the collision and the impact parameter, i.e., the point of impact on projected onto the plane . The outgoing data is analogously defined as , where is the velocity of the particle after the collision and the exit parameter, cf. Figure 1. Since we assume the scattering map is spherically symmetric, it is sufficent to define for for , where denotes the unit vector in the th coordinate direction. Any spherically symmetric scattering map (2.2) which preserves angular momentum is thus uniquely determined by
(2.3) 
where is called the scattering angle.
To satisfy the conditions of [21], we assume in the statements of Theorems 1.1 and 1.2 that one of the following hypotheses is true (cf. Fig. 2):

is strictly decreasing with and ;

is strictly increasing with and .
This assumption holds for a large class of scattering potentials, including muffintin Coulomb potentials, cf. [21]. In the case of hardsphere scatterers we have and hence Hypothesis (A) holds. For later use we define the minimal deflection angle by
(2.4) 
Note that for more general impact parameters of the form
(2.5) 
we have (by spherical symmetry)
(2.6) 
with the matrix
(2.7) 
where
(2.8) 
More explicitly,
(2.9) 
We extend the definition of to by setting for even and for odd. This choice ensures that .
For the case of general initial data , assume and are chosen so that
(2.10) 
Then
(2.11) 
We use an inductive argument to work out the velocity after the th collision, as well as the impact and exit parameters and of the th collision.
Lemma 2.1.
Fix and so that , and denote by , , the sequence of velocities, impact and exit parameters of a given particle trajectory. Then there is a unique sequence in such that for all
(2.12) 
where
(2.13) 
Proof.
We proceed by induction. We have and thus . We define by
(2.14) 
Then the assumption (2.10) is satisfied and (2.11) yields
(2.15) 
which proves the case . Let us therefore assume the statement is true for . By the induction hypothesis, we have . Note that implies , and define by
(2.16) 
Therefore (2.10) holds with , , and we can apply (2.11):
(2.17) 
where . This completes the proof. ∎
3. The BoltzmannGrad limit
We now recall the results of [20, 21] that are relevant to our investigation. Define the Markov chain
(3.1) 
on the state space with transition probability
(3.2) 
We will discuss the transition kernel in detail in Section 5. At this point, it sufficies to note that it is independent of and symmetric, i.e. . It is also independent of the choice of the scattering angle , the lattice and the initial particle distribution [20]. (Note that is related to the kernel studied in [20, 21, 22] by .) Let
(3.3) 
(3.4) 
with the mean free path length . Both and define probability densities on with respect to . The first fact follows from the symmetry of the transition kernel, and the second from the relation
(3.5) 
Suppose in the following that the sequence of random variables
(3.6) 
is given by the Markov chain (3.1), where has density either (for the continuous time setting) or (for the discrete time setting). The relation (3.4) between the two reflects the fact that the continuous time Markov process is a suspension flow over the discrete time process, where the particle moves with unit speed between consecutive collisions; see [21, Sect. 6] for more details.
We assume in the following that is a function which satisfies and which is smooth when restricted to . An example is
(3.7) 
where , and , .
For , define the following random variables:
(3.8) 
(3.9) 
(3.10) 
(3.11) 
(3.12) 
Theorem 3.1 ([21]).
(i) Under the hypotheses of Theorem 1.1, for any ,
(3.13) 
as , where the random variable has density .
(ii) Under the hypotheses of Theorem 1.2, for any ,
(3.14) 
as , where the random variable has density .
The main part of this paper is devoted to the proof of the following superdiffusive central limit theorem for the processes and , which in turn implies Theorems 1.1 and 1.2. We will only assume that the random variable is such that the marginal distribution of is absolutely continuous on with respect to Lebesgue measure; there is no further assumption on the distribution of . This hypothesis is satisfied for with density , since
(3.15) 
That is, the marginal distribution of is uniform on . We will later see that with density also complies with the above hypothesis (cf. Proposition 10.1). The processes and are independent of and , respectively, and we will in the following fix . Also, the required assumptions on the scattering angle are significantly weaker than in the previous theorems.
Theorem 3.2.
Let , and assume that the marginal distribution of is absolutely continuous. Assume is measurable, so that
(3.16) 
Then (i)
(3.17) 
and (ii)
(3.18) 
Statement (ii) in Theorem 3.1 generalizes to the convergence of the random curve (3.1) obtained by linearly interpolating [21, Theorem 1.1]. That is, under the conditions of Theorem 1.4, for and arbitrary fixed ,
(3.19) 
where the rescaled discretetime limiting process is defined by
(3.20) 
and
(3.21) 
denotes the linear interpolation of the discrete time displacements We will prove in Section 12 that the converges to in finitedimensional distribution. The last missing ingredient in the proof of Theorem 1.4 is then the tightness of the probability measures associated with the sequence of random curves in , which is established in Section 13. Theorem 1.3 follows from Theorem 1.4 via estimates presented in Section 14.
It is interesting to compare the above results with the case of a random, rather than periodic, scatterer configuration, where the scatterers are placed at the points of a fixed realisation of a Poisson process in . In the case of fixed scattering radius there is, to the best of our knowledge, no proof of a central limit theorem even in dimension . In the BoltzmannGrad limit, however, the work of Gallavotti [16], Spohn [26] and Boldrighini, Bunimovich and Sinai [7] shows that we have an analogue of Theorem 3.1, where the limit random flight process is governed by the linear Boltzmann equation. In this setting, (3.6) is a sequence of independent random variables, where has density and is uniformly distributed in . Routine techniques [25] show that in this case the central limit theorem holds for with a standard normalisation, and for with a normalisation.
4. Outline of the proof of Theorem 3.2
We will now outline the central arguments in the proof of Theorem 3.2 (ii) for discrete time by reducing the statement to four main lemmas, whose proof is given in Section 9. The continuoustime case (i) follows from (ii) via technical estimates supplied in Section 11. We will assume from now on that has density , and discuss the generalisation to more general distributions in Section 10. We note that for uniformly distributed in ,
(4.1) 
and it is therefore equivalent to consider instead of (3.6) the Markov chain
(4.2) 
with the same transition probability (3.2), uniformly distributed in and . The sequence
(4.3) 
with as defined above, is itself generated by a Markov chain on the state space with transition probability
(4.4) 
where
(4.5) 
The objective is to prove a central limit theorem of sums of the random variables . The first observation is that these are of course not independent. If we, however, condition on the sequence , then the are deterministic, and is a sequence of independent (but not identically distributed) random variables,
(4.6) 
The plan is now to apply the Lindeberg central limit theorem to the sum of independent random variables, , conditioned on .
To this end we first truncate by defining the random variable
(4.7) 
with
(4.8) 
for some fixed . The following lemma tells us that it is sufficient to prove Theorem 3.2 (ii) for instead of .
Lemma 4.1.
We have
(4.9) 
almost surely.
To prove the central limit theorem for , we center by setting
(4.10) 
with the conditional expectation
(4.11) 
where and
(4.12) 
Let
(4.13) 
The following lemma shows that and are close relative to .
Lemma 4.2.
The sequence of random variables
(4.14) 
is tight if , and
(4.15) 
is tight if .
It is therefore sufficient to prove Theorem 3.2 (ii) for in place of . This will be achieved by applying the Lindeberg central limit theorem to the conditional sum as aluded to above. We begin by estimating the conditional variance. Set
(4.16) 
with
(4.17) 
Lemma 4.3.
There is a constant such that, for ,
(4.18) 
By taking the trace in (4.18), we have in particular
(4.19) 
for
(4.20) 
Recall that convergence in probability is defined as for any .
The next lemma verifies the Lindeberg conditions for random .
Lemma 4.4.
For any fixed ,
(4.21) 
as .
Given these lemmas, let us now conclude the proof of the fact that
(4.22) 
By Chebyshev’s inequality we have, for any ,
(4.23) 
and thus, for any ,
(4.24) 
By (4.19), the second term on the right hand side of (4.24) converges to as , if we choose , say. So (4.24) implies that the sequence of random variables is tight. By the HellyProkhorov theorem, there is an infinite subset so that converges in distribution along to some limit . Assume for a contradiction that is not distributed according to . The BorelCantelli lemma implies that there is an infinite subset , so that in the statements of Lemmas 4.3 and 4.4 we have almostsure convergence along :
(4.25) 
(4.26) 
and
(4.27) 
The hypotheses of the Lindeberg central limit theorem are met, and we infer that for . (We use the Lindeberg theorem for triangular arrays of independent random variables, since we have verified the Lindeberg conditions only along a subsequence.) This, however, contradicts our assumption that is not normal, and hence is indeed the unique limit point of any converging subsequence. This in turn implies that every sequence converges, and therefore completes the proof of (4.22). In view of Lemmas 4.1 and 4.2, this implies Theorem 3.2 (ii) (still under the assumption that has density ).
Let us briefly describe the further contents of this paper. In Section 5 we recall the basic properties of the transition kernel from [22]. Section 6 establishes key estimates for the moments , and introduced above. In Sections 7 and 8 we prove spectral gap estimates and exponential mixing for the discrete time Markov process defined in (4.4). The estimates from Sections 6–8 are the main input in the proof of Lemmas 4.1–4.4, which is given in Section 9. In Section 10 we show that the discretetime statement in Theorem 3.2 (ii) holds for more general initial distributions than