Wind-tree Model Research Articles

Normal and anomalous diffusion in the disordered wind-tree model.

Ehrenfests' wind-tree model (EWTM) refers to a two-dimensional system where noninteracting point tracer particles move through a random arrangement of overlapping or nonoverlapping square-shaped scatterers. Here, extensive event-driven molecular dynamics simulations of the EWTM at different reduced scatterer densities ρ are presented. For nonoverlapping scatterers, the asymptotic motion of the tracer particles is diffusive. We compare their diffusion coefficient D, as obtained from the simulation, with that predicted by kinetic theory where D^{-1} is expanded up to the second order in the scatterer density. While at low density quantitative agreement between theory and simulation is found, we show that beyond the low-density regime deviations to the theory are associated with the emergence of a maximum in the non-Gaussian parameter at intermediate times. For the case of overlapping scatterers, in agreement with a theoretical prediction, the asymptotic motion of the tracer particles is subdiffusive, i.e., the mean-squared displacement at long times t grows like t^{1-2ρ/3}. We propose a model of the van Hove correlation function that describes the density dependence of the tracer particles' asymptotic subdiffusive transport on a quantitative level.

Ergodicity, mixing, Ratner's properties and disjointness for classical flows: On the research of Corinna Ulcigrai

<p style='text-indent:20px;'>We present and discuss C. Ulcigrai's results concerning mixing properties of locally Hamiltonian flows, spectral properties of smooth time changes of horocycle flows together with their Möbius orthogonality and the ergodicity problems of directional flows in the wind tree model of Ehrenfest.</p>

Invariance Principle for the Random Wind-Tree Process

Consider a point particle moving through a Poisson distributed array of cubes all oriented along the axes—the random wind-tree model introduced in Ehrenfest–Ehrenfest (1912) as reported by Ehrenfest, Ehrenfest (Begriffliche Grundlagen der statistischen Auffassung in der Mechanik Encykl. d. Math. Wissensch. IV 2 II, Heft 6, 90 S (1912) (Translated:) The conceptual foundations of the statistical approach in mechanics. Dover Books on Physics, 1912). We show that in the joint Boltzmann–Grad and diffusive limit this process satisfies an invariance principle. That is, the process converges in distribution to Brownian motion in a particular scaling limit. In a previous paper (2020) (Lutsko, Tóth in Commun. Math. Phys. 379:589–632, 2020) the authors used a novel coupling method to prove the same statement for the random Lorentz gas with spherical scatterers. In this paper we show that, despite the change in dynamics, a similar strategy with some modification can be used to prove the invariance principle for the random wind-tree model. The key differences from our previous work are that the individual path segments of the underlying Markov process are no longer fully independent and the geometry of recollisions is simpler.

Diffusion Rate of Windtree Models and Lyapunov Exponents

Consider a windtree model with one or several right-angled obstacles placed periodically on the plan. We show that their diffusion rate is associated to the largest Lyapunov exponent of some quadratic differentials stratum, and exhibit a general strategy to associate to a windtree model a Lyapunov exponent corresponding to its diffusion rate. This results enables us to compute numerically these diffusion rates, and to observe via computer experiments its behaviour according to the shape of the obstacles.

Ehrenfests’ Wind–Tree Model is Dynamically Richer than the Lorentz Gas

We consider a physical Ehrenfests' Wind-Tree model where a moving particle is a hard ball rather than (mathematical) point particle. We demonstrate that a physical periodic Wind-Tree model is dynamically richer than a physical or mathematical periodic Lorentz gas. Namely, the physical Wind-Tree model may have diffusive behavior as the Lorentz gas does, but it has more superdiffusive regimes than the Lorentz gas. The new superdiffusive regime where the diffusion coefficient D(t)~(ln t)^2 of dynamics seems to be never observed before in any model.

Quantitative error term in the counting problem on Veech wind-tree models

We study periodic wind-tree models, billiards in the plane endowed with $\mathbb{Z}^2$-periodically located identical connected symmetric right-angled obstacles. We exhibit effective asymptotic formulas for the number of (isotopy classes of) periodic billiard trajectories (up to $\mathbb{Z}^2$-translations) on Veech wind-tree billiards, that is, wind-tree billiards whose underlying compact translation surfaces are Veech surfaces. We show that the error term depends on spectral properties of the Veech group and give explicit estimates in the case when obstacles are squares of side length $1/2$.

Ergodic properties of the ideal gas model for infinite billiards

In this paper we study ergodic properties of the Poisson suspension (the ideal gas model) of the billiard flow (bt)t∈R on the plane with a Λ-periodic pattern (Λ⊂R2 is a lattice) of polygonal scatterers. We prove that if the billiard table is additionally rational then for a.e. direction θ∈S1 the Poisson suspension of the directional billiard flow (btθ)t∈R is weakly mixing. This gives the weak mixing of the Poisson suspension of (bt)t∈R. We also show that for a certain class of such rational billiards (including the periodic version of the classical wind-tree model) the Poisson suspension of (btθ)t∈R is not mixing for a.e. θ∈S1.

A Non-varying Phenomenon with an Application to the Wind-Tree Model

Abstract We exhibit a non-varying phenomenon for the counting problem of cylinders, weighted by their area, passing through two marked (regular) Weierstrass points of a translation surface in a hyperelliptic connected component $\mathcal{H}^{hyp}(2g-2)$ or $\mathcal{H}^{hyp}(g-1,g-1)$, $g> 1$. As an application, we obtain the non-varying phenomenon for the counting problem of (weighted) periodic trajectories on the Ehrenfest wind-tree model, a billiard in the plane endowed with $\mathbb{Z}^2$-periodically located identical rectangular obstacles.

Recurrence and non-ergodicity in generalized wind-tree models

In this paper, we consider generalized wind-tree models and Zd-covers over compact translation surfaces. Under suitable hypothesis, we prove the recurrence of the linear flow in a generic direction and the non-ergodicity of Lebesgue measure.

Counting problem on wind-tree models

We study periodic wind-tree models, billiards in the plane endowed with $\mathbb{Z}^2$-periodically located identical connected symmetric right-angled obstacles. We show asymptotic formulas for the number of (isotopy classes of) closed billiard trajectories (up to $\mathbb{Z}^2$-translations) on the wind-tree billiard. We also compute explicitly the associated Siegel-Veech constant for generic wind-tree billiards depending on the number of corners on the obstacle.

Infinite Ergodic Index of the Ehrenfest Wind-Tree Model

The set of all possible configurations of the Ehrenfest wind-tree model endowed with the Hausdorff topology is a compact metric space. For a typical configuration we show that the wind-tree dynamics has infinite ergodic index in almost every direction. In particular some ergodic theorems can be applied to show that if we start with a large number of initially parallel particles their directions decorrelate as the dynamics evolve, answering the question posed by the Ehrenfests.

Recurrence for the wind-tree model

In this paper, we give a geometric criterion ensuring the recurrence of the vertical flow on Zd-covers of compact translation surfaces (d≥2). We prove that the linear flow in the wind-tree model is recurrent for every pair of parameters and almost every direction.

Ergodicity of the Ehrenfest wind–tree model

We consider aperiodic wind–tree models, and show that, for a generic (in the sense of Baire) configuration, the wind–tree dynamics is ergodic in almost every direction.

Minimality of the Ehrenfest wind-tree model

We consider aperiodic wind-tree models, and show that for a generic (in the sense of Baire) configuration the wind-tree dynamics is minimal in almost all directions, and has a dense set of periodic points.

Pseudo-Anosov eigenfoliations on Panov planes

We study dynamical properties of direction foliations on the complex plane pulled back from direction foliations on a half-translation torus $T$, i.e., a torus equipped with a strict and integrable quadratic differential. If the torus $T$ admits a pseudo-Anosov map we give a homological criterion for the appearance of dense leaves and leaves with bounded deviation on the universal covering of $T$, called Panov plane. Our result generalizes Dmitri Panov's explicit construction of dense leaves for certain arithmetic half-translation tori [33]. Certain Panov planes are related to the polygonal table of the periodic wind-tree model. In fact, we show that the dynamics on periodic wind-tree billiards can be converted to the dynamics on a pair of singular planes. <br> Possible strategies to generalize our main dynamical result to larger sets of directions are discussed. Particularly we include recent results of Frączek and Ulcigrai [17, 18] and Delecroix [6] for the wind-tree model. Implicitly Panov planes appear in Frączek and Schmoll [15], where the authors consider Eaton Lens distributions.

Diffusion for the periodic wind-tree model

The periodic wind-tree model is an infinite billiard in the plane with identical rectangular scatterers placed at each integer point. We prove that independently of the size of scatters and generically with respect to the angle, the polynomial diffusion rate in this billiard is 2/3.

Non-ergodic $\mathbb{Z}$ -periodic billiards and infinite translation surfaces

We give a criterion which proves non-ergodicity for certain infinite periodic billiards and directional flows on $\mathbb{Z}$ -periodic translation surfaces. Our criterion applies in particular to a billiard in an infinite band with periodically spaced vertical barriers and to the Ehrenfest wind-tree model, which is a planar billiard with a $\mathbb{Z}^{2}$ -periodic array of rectangular obstacles. We prove that, in these two examples, both for a full measure set of parameters of the billiard tables and for tables with rational parameters, for almost every direction the corresponding directional billiard flow is not ergodic and has uncountably many ergodic components. As another application, we show that for any recurrent $\mathbb{Z}$ -cover of a square tiled surface of genus two the directional flow is not ergodic and has no invariant sets of finite measure for a full measure set of directions. In the language of essential values, we prove that the skew-products which arise as Poincaré maps of the above systems are associated to non-regular $\mathbb{Z}$ -valued cocycles for interval exchange transformations.

Divergent trajectories in the periodic wind-tree model

The periodic wind-tree model is a family T(a,b) of billiards in the plane in which identical rectangular scatterers of size axb are disposed at each integer point. It was proven by P. Hubert, S. Leli\`evre and S. Troubetzkoy (arXiv:0912.2891v1) that for a residual set of parameters (a,b) the billiard flow in T(a,b) is recurrent in almost every direction. We prove that for many parameters (a,b) there exists a set S of angles of positive Hausdorff dimension such that every billiard trajectory in T(a,b) with initial angle in S is self-avoiding. In particular, the flow in a direction of S is divergent.

Capillary Floating and the Billiard Ball Problem

We study the recurrence and ergodicity for the billiard on noncompact polygonal surfaces with a free, cocompact action of $\Z$ or $\Z^2$. In the $\Z$-periodic case, we establish criteria for recurrence. In the more difficult $\Z^2$-periodic case, we establish some general results. For a particular family of $\Z^2$-periodic polygonal surfaces, known in the physics literature as the wind-tree model, assuming certain restrictions of geometric nature, we obtain the ergodic decomposition of directional billiard dynamics for a dense, countable set of directions. This is a consequence of our results on the ergodicity of $\ZZ$-valued cocycles over irrational rotations.

Typical Recurrence for the Ehrenfest Wind-Tree Model

We show that the typical wind-tree model, in the sense of Baire, is recurrent and has a dense set of periodic orbits. The recurrence result also holds for the Lorentz gas: the typical Lorentz gas, in the sense of Baire, is recurrent. These Lorentz gases need not be of finite horizon!