Skip to main content
SpringerLink
Log in

Invariance Principle for the Random Wind-Tree Process

  • Published:
Annales Henri Poincaré Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Avila, A., Hubert, P.: Recurrence for the wind-tree model. Ann. I. H, Poincaré - AN 37, 1–11 (2017)

  2. Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)

    MATH  Google Scholar 

  3. Boldrighini, C., Bunimovich, L.A., Sinai, Y.G.: On the Boltzmann equation for the Lorentz gas. J. Stat. Phys. 32, 477–501 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  4. Delecroix, V.: Divergent trajectories in the periodic wind-tree model. J. Modern Dyn. 7, 1–29 (2013)

  5. Delecroix, V., Hubert, P., Lelièvre, S.: Diffusion for the wind tree model. Ann. Sci. Ec. Norm. Supér. (4)47(6), 1085–1110 (2014)

    Article  MathSciNet  Google Scholar 

  6. Ehrenfest, P., Ehrenfest, T.: 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 9780486662503 (1959)

  7. Fraczek, K., Ulcigrai, C.: Non-ergodic \(z\)-periodic billiards and infinite translation surfaces. Invent. Math. 197(2), 241–298 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  8. Gallavotti, G.: Divergencies and the approach to equilibrium in the Lorentz and the wind-tree models. Phys. Rev. 185, 308–322 (1969)

    Article  ADS  Google Scholar 

  9. Gallavotti, G.: Rigorous theory of the Boltzmann equation in the Lorentz gas. Nota Int. Univ. di Roma 358, 21 (1970)

    Google Scholar 

  10. Hardy, J., Weber, J.: Diffusion in a periodic wind-tree model. J. Math. Phys. 21, 1802 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  11. Hubert, P., Lelièvre, S., Troubetzkoy, S.: The Ehrenfest wind-tree model: periodic directions, recurrence, diffusion. J. Reine Angew. Math. 656, 223–244 (2011)

    MathSciNet  MATH  Google Scholar 

  12. Lorentz, H.A. : The motion of electrons in metallic bodies. In: Proc. Amstredam Acad. 7: 438, 585, 604 (1905)

  13. Lutsko, C., Tóth, B.: Invariance principle for the random Lorentz gas - beyond the Boltzmann-Grad limit. Commun. Math. Phys. 379(2), 589–632 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  14. Málaga Sabogal, A., Troubetzkoy, S.: Infinite ergodic index of the Ehrenfest wind-tree model: Communications in Mathematical Physics 358, 995–1006 (2018)

  15. Marklof, J.: The low-density limit of the Lorentz gas: periodic, aperiodic and random. In: Proceedings of the International Congress of Mathematicians – 2014 Seoul Vol. 3, 623-646, Kyung Moon Sa, Seoul, (2014)

  16. Marklof, J., Strömbergsson, A.: Kinetic theory for the low density Lorentz gas. arXiv:1910.04982 [math.DS] (2019)

  17. Spohn, H.: The Lorentz process converges to a random flight process. Commun. Math. Phys. 60, 277–290 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  18. Tabachnikov, S.: Billiards. Panoramas et Synthèses, Société mathématique de France 1 (1995)

Download references

Acknowledgements

The work of BT was supported by EPSRC (UK) Fellowship EP/P003656/1 and by NKFI (HU) K-129170. CL was supported by EPSRC Studentship EP/N509619/1 1793795.

Author information

Authors and Affiliations

  1. School of Mathematics, University of Bristol, Bristol, BS8 1TW, United Kingdom

    Christopher Lutsko

  2. Rényi Institute, Budapest, Hungary

    Bálint Tóth

Authors
  1. Christopher Lutsko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bálint Tóth
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christopher Lutsko.

Additional information

Communicated by Christian Maes.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lutsko, C., Tóth, B. Invariance Principle for the Random Wind-Tree Process. Ann. Henri Poincaré 22, 3357–3389 (2021). https://doi.org/10.1007/s00023-021-01106-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00023-021-01106-4

Mathematics Subject Classification

Search

Navigation