A few weeks ago, I was invited by my friend Jairo Bochi to give a “general audience like” talk (that I’ll deliver today) at UFRJ (Brazil) in a Dynamical Systems seminar called EDAI. After thinking a bit, I decided to discuss a recent beautiful theorem of V. Delecroix, P. Hubert and S. Lelièvre on the diffusion rates for the Ehrenfest wind-tree model of Lorenz gases. Here, my choice of theme was motivated by the following two facts:
- this theorem has some roots in Physics (more precisely, Statistical Mechanics) and
- its proof has a lot to do with recent advances in the study of the Lyapunov exponents of the Kontsevich-Zorich cocycle (a subject that I’m particularly interested in).
So, while I was preparing these slides here (written in Portuguese), I thought that it could be a nice idea to publish a sort of “extended version” of the slides in this blog. The outcome of this is the text below the fold. I hope you’ll enjoy your reading as much as I enjoyed preparing these notes!
–Ehrenfest wind-tree model–
In 1912, Paul and Tatiana Ehrenfest introduced in their article “Begriffliche Grundlagen der statistischen Auffassung in der Mechanik” (a review on Statiscal Mechanics later translated as The conceptual Foundations of the Statistical Approach in Mechanics) the following model (called “wind-tree model”) of Lorenz gases: one considers a particle moving in a billiard table obtained by removing from the plane a certain number of rectangular obstacles.
In 1980, J. Hardy and J. Weber studied a periodic version of the wind-tree model where rectangular obstacles of dimensions , , are disposed along the integral lattice (i.e., the centers of these rectangles are exactly the points of the plane with integral coordinates).
The figures below (extracted from Vincent Delecroix’s Ph.D. thesis) show some examples of billiard trajectories in periodic wind-tree models.
The meaning of the colors in this figure is the following. In the left side of this picture, our trajectory starts with angle (with the horizontal direction), and the initial piece is colored in blue. Then, it collides with a rectangle, and the new angle is , and the corresponding piece of trajectory is colored in yellow. After that, we have a collision leading to a green piece of trajectory with angle , and finally a collision leading to a red piece of trajectory with angle .
The script used to produce the figure above is available at Vincent Delecroix’s webpage.
Denote by the translation flow in a periodic wind-tree model in the direction starting at . In their article (quoted above), J. Hardy and J. Weber showed that the periodic wind-tree model exhibits a diffusion of (i.e., for a.e. , one has ) along special directions , , corresponding to generalized diagonals (e.g., when , is such a direction). Roughly speaking, the proof of the diffusion result of J. Hardy and J. Weber is based on the observation that the translation flow along generalized diagonals of periodic wind-tree models can be interpreted as a particular type of skew-product over a rotation of the circle (and this kind of dynamical system can be reasonably analyzed with direct calculations). Moreover, it was implicitly conjectured that one should expect “abnormal diffusion” along typical directions , that is,
Here, the “justification” of the word “abnormal” comes by comparison with Brownian motion and/or central limit theorem: once we know that the diffusion is “sublinear” (maybe after removing the “average”), the “natural” scaling factor to converge (maybe to a normal distribution) is , i.e., a “normal” diffusion would correspond to
Remark 1 In the case of translation flows, a theorem of S. Kerckhoff, H. Masur and J. Smillie ensures that the diffusion is “always” sublinear, i.e.,
for almost every and .
In this direction, the following (very recent) theorem of V. Delecroix, P. Hubert and S. Lelièvre confirms the “abnormal” diffusion conjecture:
Theorem 1 (Delecroix, Hubert, Lelièvre (2011)) For the periodic wind-tree model, one has
in the following cases:
- (a) for a.e. , , and any with infinite -trajectory;
- (b) for , for a.e. , and any with infinite -trajectory;
- (c) for , , with and , for a.e. , and any with infinite -trajectory.
Our main goal here is the presentation of a sketch of proof of this theorem. In particular, we will see how the value emerges in the result above: in a nutshell, it is a Lyapunov exponent of the Kontsevich-Zorich cocycle over a -invariant locus of genus 5 Riemann surfaces.
Remark 2 In fact, Vincent Delecroix found recently that the speed of diffusion may be faster than 2/3 for other wind-tree models (namely, -periodic wind-tree models where is an appropriately chosen lattice in ). We will say a few words on this by the end of this post.
The organization of the subsequent discussion is as follows. In the next section, we briefly recall the so-called Katok-Zemlyakov construction of translation surfaces associated to rational billiards. Then, in a subsequent section, we reduce the study of diffusion speed to the study of algebraic intersections of pieces of trajectories of with certain -valued cocycles. Afterwards, a section will be dedicated to relate these algebraic intersections with the Lyapunov exponents of the Kontsevich-Zorich cocycle (by following the works of A. Zorich and G. Forni). At this point, the sketch of proof of Delecroix-Hubert-Lelièvre theorem will be completed by computing the relevant Lyapunov exponents (and deriving the value ). Finally, the last section will consist in a few words on Remark 2 above.
Given a rational polygon , that is, a polygon whose inner angles are all rational multiples of , it is not hard to see that the billiard flow in is equivalent to a translation flow in a translation surface (see this link here for definitions and more details on these notions) obtained by reflecting the sides of the polygon every time a trajectory hits the boundary of . In other words, instead of changing of direction by the reflection law in the billiard , we keep the same direction but reflect the polygon in order to be able to continue the trajectory. Below, the reader can see a figure (extracted from slides of a talk of Corinna Ulcigrai) where this unfolding process is illustrated in the case of a square:
In the literature, this unfolding procedure was introduced by R. Fox and R. Keshner (in 1936) and rediscovered (independently) by A. Katok and A. Zemlyakov (in 1975), and it is popularly called Katok-Zemlyakov construction.
In the particular case of the figure above, we get a torus as the resulting translation surface of the Katok-Zemlyakov construction applied to the billiard in the unit square.
We observe that the rationality condition above was imposed on the polygon only to get that the associated translation surface is compact (as the group generated by the reflections along the sides of is finite when the inner angles belong to ). However, the Katok-Zemlyakov construction can also be applied to the non-compact billiard table of the -periodic wind-tree model with parameters , but the resulting translation surface is not easy to visualize.
We overcome this visualization difficulty as follows. The figure below shows a fundamental domain in :
Here, the meaning of the vectors and near the boundaries of the fundamental domain is the following. We cover with countably many copies of indexed by in the natural way. Then, any billiard trajectory in may be coded by a billiard trajectory in and a sequence of integers determining the copies of the trajectory is passing by. In other words, every time a trajectory in hits the boundary, we use the corresponding vector (one of the vectors and ) to indicate that we’re changing from a copy of of index to the copy of of index . Using technical jargon, this means that we can see the billiard on as a skew-product of the billiard on by a -valued cocycle.
In this context, we can apply the Katok-Zemlyakov construction to . The resulting translation surface is indicated below.
It follows that the translation surface obtained from the Katok-Zemlyakov construction applied to is a -cover of (with the -cocycle indicated in the picture above).
Remark 3 A “more natural” fundamental domain for the billiard in is showed in the left side of the picture below:
Concerning the (flat) geometry of the translation surface , one can check that it has distinct conic singularities with total angle , that is, the natural Abelian differential on has 4 double zeros, i.e., . By Gauss-Bonnet theorem, is a Riemann surface of genus 5.
In resume, the billiard flow in direction on is dynamically equivalent to a skew-product given by a -cocycle over the translation flow in direction on the genus 5 translation surface .
–Speed of diffusion and algebraic intersections of cycles–
The speed of diffusion of (with ) can be studied as follows.
In the genus 5 surface , denote by the segment of orbit and consider the closed segment obtained from by “closing” its endpoints and with a segment of bounded length.
Next, we introduce the cycle
where , , are the cycles indicated in Figure 5.
Since , it makes sense to consider the algebraic intersection . In this notation, we have the following fact:
Lemma 2 It holds . In particular,
Proof: The second statement follows from the first one, and the latter follows by definitions and the fact that the fundamental domain has diameter .
From this lemma, we get:
In other words, the study of speed of diffusion is equivalent to the study of algebraic intersections of certain cycles in .
Remark 4 The algebraic intersections correspond to “special Birkhoff sums”: for sufficiently small times and it stays like that until hits the “boundary” of , say at a time ; at this point, for sufficiently close to , where is a vector of the form or (depending on the boundary cycle we landed), and it stays like that until hits again the “boundary” of , say at a time ; at this point, for sufficiently close to , where is a vector of the form or (depending on the boundary cycle we landed); and so on ad infinitum.
–Algebraic intersections, Teichmüller flow and Kontsevich-Zorich cocycle–
Following the works of Anton Zorich (1994 and 1997), and Giovanni Forni (1997 and 2002), we will try to understand the algebraic intersections by means of the so-called Kontsevich-Zorich cocycle over the Teichmüller flow.
Roughly speaking, the basic idea is: is a very long and complicated segment (of length ) in , so that a direct study might be complicated; in order to overcome this difficulty, we make shorter, by first rotating (if necessary) so that one can suppose that , and then applying the diagonal matrix to ; since is a vertical segment of length on , by applying to , one gets a vertical segment of length on .
In particular, by taking (i.e., ), we have that is a vertical segment of length . The picture below shows the effect of on a translation surface formed by 4 “zippered rectangles”:
In the literature, the action of on translation surfaces is called Teichmüller flow. In this language, we see that the Teichmüller flow can be used to shorten pieces of trajectories of the translation flow in translation surfaces, or in other words,
The Teichmüller flow is a nice renormalization procedure on translation flows.
Of course, there is “price” to pay: our unit length segment doesn’t live anymore in the surface , but in the distorted surface . At this point, we have the impression that the use of Teichmüller flow only pushes the difficulty from one place to another, but this is not quite true: it is known that the Teichmüller flow has nice recurrence properties in the moduli space of translation surfaces, that is, for infinitely many times , is “essentially” the same surface as in the moduli space, i.e., and are very similar surfaces up to some cutting and pasting operations. The figure below (extracted from A. Zorich’s survey) illustrates some “cutting and pasting operations” in a surface derived from a translation surface by applying the Teichmüller flow:
The cutting and pasting operations on a surface of genus correspond to the action of the so-called mapping class-group of isotopy classes of orientation-preserving diffeomorphisms of : indeed, these cutting and pasting operations are simply changes on the “names” of homology classes in given by the natural (symplectic) action on homology of certain diffeomorphisms.
Going back to our initial problem of studying algebraic intersections , we saw that the action of on allows us to think of ( and also) (to some extent) as a segment of length for in a surface essentially “equal” to (at least for some infinite sequence of times ). However, since the equality in the moduli space of translation surfaces only means that is equal to up to some element of the mapping-class group, one may change into another cycle when using to “cut and paste” into .
In other words, because the algebraic intersection is preserved by the natural action on homology of the diffeomorphism , one has
Remark 5 Strictly speaking, the previous equality is not exactly true: while the Teichmüller flow makes shorter, since is obtained from by “closing” its endpoints with arbitrarysegments of bounded length, this introduces some little technical difficulties. However, it is possible to show that the potential discrepancies introduced by these “closing” procedures don’t affect the study of speed of diffusion, that is, it holds
Here, we used the “time change” .
In the literature, the family of linear operators (matrices) on the first homology group induced by the action of the elements of the mapping-class group used to convert into a surface “essentially” equal to is called the Kontsevich-Zorich cocycle over the Teichmüller flow. Keeping this in mind, one sees that the expression measures the (exponential) rate of growth of , i.e., the study of speed of diffusion is equivalent to the study of the exponential rate of growth of the Kontsevich-Zorich cocycle applied to the particular homology cycle .
In resume, the problem of studying algebraic intersections is equivalent to the question of determining the Lyapunov exponents of the Kontsevich-Zorich (KZ) cocycle. Again, this seems to lead us nowhere, but this is not quite true: Lyapunov exponents of KZ cocycle received a lot of attention recently, and, as we’re going to see in the next section, we can compute them in several situations (specially in the presence of symmetric surfaces such as ).
–End of “proof” of Delecroix-Hubert-Lelièvre theorem–
As one can check in Figure 5, has a group of symmetries isomorphic to the Klein group . This group of symmetries is generated by a natural “horizontal translation” and a natural “vertical translation” such that (i.e., and are involutions). In particular, we can define , resp. , the subspaces of , resp. , invariant homology classes, and , resp. , the subspaces of , resp. , anti-invariant homology classes. Using these subspaces, one can introduce
By definition, we have . Moreover, since it is possible to verify that the action on homology of and commutes with the Kontsevich-Zorich cocycle (“ and acts by pre-composition with translation charts while KZ cocycle acts by post-composition with translation charts”), the KZ cocycle preserves the susbpaces , , and, hence, can be diagonalized by blocks. Therefore, Lyapunov exponents of are the ones coming from the blocks , , and . Moreover, since the symplectic intersection form is preserved, the Lyapunov spectrum on each block is symmetric: if is a Lyapunov exponent, then is also a Lyapunov exponent. Thus, the Lyapunov spectrum on a block is determined by the list of non-negative Lyapunov exponents. In the sequel, when talking about Lyapunov exponents, we’ll care exclusively about the non-negativeones.
We start with the block . By definition, the quotient is (shown in Figure 4). The translation surface has an unique conical singularity with total angle , that is, (i.e., the natural Abelian differential on has a single zero of order 2). Moreover, by definition, is canonically identified with the homology group , and, under this identification, corresponds to the Kontsevich-Zorich cocycle associated to the -orbit of .
- (a) dense in for a.e. ;
- (b) closed (Veech non-arithmetic) for , , but , ;
- (c) closed (Veech arithmetic) for .
In particular, the statement of Delecroix-Hubert-Lelièvre theorem is justified by this classification result.
Moreover, in the case of , the Lyapunov exponents of the Kontsevich-Zorich cocycle were computed by M. Bainbridge and, more recently, by A. Eskin, M. Kontsevich and A. Zorich (using a far-reaching formula to appear in a work still in preparation). The outcome of their work is the fact that the Kontsevich-Zorich cocycle on , or equivalently , has Lyapunov exponents and .
Next, we analyze the blocks and . The surfaces and are hyperelliptic Riemann surfaces of genus in the odd spin connected component of translation surfaces with two double zeroes (see this post here and references therein for more details on connected components of strata ), that is, and belong to the hyperelliptic locus of . In this case, by the aforementioned work of A. Eskin, M. Kontsevich and A. Zorich (see also this recent paper of D. Chen and M. Möller), one gets that the sum of the 3 non-negative Lyapunov exponents of KZ cocycle over these surfaces is . On the other hand, by definition, , resp. , is canonically identified to , resp. , and , resp. , corresponds to the KZ cocycle over , resp. .
In particular, the sum of the non-negative Lyapunov exponents of , resp. is . However, we already saw that contributes with the Lyapunov exponents and , so that , resp. , contributes with a Lyapunov exponent .
Finally, we analyze the block . We consider the surface . It belongs to the hyperelliptic connected component of the stratum genus 3 translation surfaces with two double zeroes. By the work of Eskin, Kontsevich and Zorich (see also the article of Chen and Möller quoted above), the sum of the 3 non-negative Lyapunov exponents of the KZ cocycle is . Here, is canonically identified with , so that contributes with a exponent (as already contributes with and ).
Therefore, the sketch of proof of Delecroix-Hubert-Lelièvre theorem will be complete as soon as we determine the relevant Lyapunov exponent controlling the growth of . By direct inspection, one sees that decomposes as a sum of and , so that the relevant exponents are (associated to and ). Moreover, it can be shown that doesn’t belong to the stable Oseledets space of the KZ cocycle (i.e., the subspace associated to the negative exponents) because is a integer cycle (so that its size can’t go to zero as ). In other words, has size about as , so that
and the argument is “complete”.
–Faster diffusion in other periodic wind-tree models–
Closing today’s post, we will say a few words on a nice argument found by V. Delecroix to construct -periodic wind-tree models () with a diffusion faster than .
The basic idea is that -periodic wind-tree models are less symmetric than their -periodic counterpart: instead of having a Klein group of symmetries, one keeps only one of the involutions , , let’s say for sake of concreteness. In this case, the quotient of by is still a genus surface, but we can’t compute all exponents individually because we don’t dispose of a nice genus 2 surface (i.e., we lose the bundle). Nevertheless, it is possible to prove that the sum of the two relevant Lyapunov exponents controlling is (recall that it was also before), and, actually, has size about as (i.e., it is who controls the rate of growth of ). However, by setting up the parameters and the lattice properly (in particular, , that is, is a square-tiled surface), one can apply a recent criterion of simplicity of Lyapunov exponents of the KZ cocycle over square-tiled surfaces developed by M. Möller, J.-C. Yoccoz and myself (see this preprint here for a statement of this criterion and its application in a very concrete case). In a certain sense, this criterion is a version for square-tiled surfaces of a simplicity result of A. Avila and M. Viana (see these papers here and here) for the KZ cocycle with respect to the Masur-Veech measure, and it is likely that we will discuss this issue in a future post. In any case, it turns out that V. Delecroix was able to check this simplicity criterion in the case of appropriately chosen and , so that one gets in this particular situation. Of course, since ,one has , and the desired result (of diffusion faster than ) follows.