Skip to main content
SpringerLink
Log in

A Functional Analytic Approach to Perturbations of the Lorentz Gas

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We present a functional analytic framework based on the spectrum of the transfer operator to study billiard maps associated with perturbations of the periodic Lorentz gas. We show that recently constructed Banach spaces for the billiard map of the classical Lorentz gas are flexible enough to admit a wide variety of perturbations, including: movements and deformations of scatterers; billiards subject to external forces; nonelastic reflections with kicks and slips at the boundaries of the scatterers; and random perturbations comprised of these and possibly other classes of maps. The spectra and spectral projections of the transfer operators are shown to vary continuously with such perturbations so that the spectral gap enjoyed by the classical billiard persists and important limit theorems follow.

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

Similar content being viewed by others

References

  1. Baladi, V.: Positive transfer operators and decay of correlations. Advanced Series in Nonlinear Dynamics 16, Singapore: World Scientific, 2000

  2. Bálint, P., Melbourne, P.: Decay of correlations for flows with unbounded roof function, including the infinite horizon planar periodic Lorentz gas. Preprint

  3. Chernov N.: Decay of correlations in dispersing billiards. J. Statt. Phys. 94, 513–556 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Chernov N.: Sinai billiards under small external forces. Ann. Henri Poincare 2, 197–236 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Chernov N.: Sinai billiards under small external forces II. Ann. H. Poincare 9, 91–107 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chernov N., Dolgopyat D.: Brownian Brownian Motion – I. Mem. Amer. Math. Soc. 198, 927 (2009)

    MathSciNet  Google Scholar 

  7. Chernov N., Dolgopyat D.: Anomalous current in periodic Lorentz gases with infinite horizon. Russ. Math. Surv. 64, 651–699 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chernov N., Eyink G., Lebowitz J., Sinai Ya.: Derivation of Ohms law in a deterministic mechanical model. Phys. Rev. Lett. 70, 2209–2212 (1993)

    Article  ADS  Google Scholar 

  9. Chernov N., Eyink G., Lebowitz J., Sinai Ya.: Steady-state electrical conduction in the periodic Lorentz gas. Commun. Math. Phys. 154, 569–601 (1993)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. Chernov N., Haskell C.: Nonuniformly hyperbolic K-systems are Bernoulli. Erg. Th. Dyn. Sys. 16, 19–44 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chernov N., Markarian R.: Chaotic Billiards. Mathematical Surveys and Monographs, 127, Providence, RI: Amer. Math. Soc, 2006

  12. Demers M.F., Liverani C.: Stability of statistical properties in two-dimensional piecewise hyperbolic maps. Trans. Amer. Math. Soc. 360(9), 4777–4814 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Demers M.F., Zhang H.-K.: Spectral analysis of the transfer operator for the Lorentz gas. J. Mod. Dyn. 5(4), 665–709 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dolgopyat D., Szász D., Varjú T.: Limit Theorems for Locally Perturbed Lorentz processes. Duke Math. J. 148, 459–499 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gallavotti G., Ornstein D.: Billiards and Bernoulli schemes. Commun. Math. Phys. 38, 83–101 (1974)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Gouëzel S., Liverani C.: Banach spaces adapted to Anosov systems. Ergc. Th. Dyn. Sys. 26(1), 189–217 (2006)

    Article  MATH  Google Scholar 

  17. Katok, A., Strelcyn, J.-M.: With the collaboration of Ledrappier, F., Przytycki, F.: Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities. Lecture Notes in Math. 1222, Berlin: Springer-Verlag, 1986

  18. Keller G., Liverani C.: Stability of the spectrum for transfer operators. Annali della Scuola Normale Sup. di Pisa, Sci. Fis. e Mat. (4) XXVIII, 141–152 (1999)

    MathSciNet  Google Scholar 

  19. Liverani, C.: Invariant measures and their properties. A functional analytic point of view. In: Dynamical Systems. Part II: Topological Geometrical and Ergodic Properties of Dynamics. Pubblicazioni della Classe di Scienze, Scuola Normale Superiore, Pisa. Centro di Ricerca Matematica “Ennio De Giorgi”: Proceedings. Pisa: Scuola Normale Superiore, 2004

  20. Markarian R., Pujals E.J., Sambarino M.: Pinball billiards with dominated splitting. Ergc. Th. Dyn. Sys. 30, 1757–1786 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Melbourne I., Nicol M.: Almost sure invariance principle for nonuniformly hyperbolic systems. Commun. Math. Phys. 260, 393–401 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Melbourne I., Nicol M.: Large deviations for nonuniformly hyperbolic systems. Trans. Amer. Math. Soc. 360, 6661–6676 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rey-Bellet L., Young L.-S.: Large deviations in nonuniformly hyperbolic dynamical systems. Erg. Th. Dyn. Sys. 28, 587–612 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sinai Ya.: Dynamical systems with elastic reflections. Ergodic. properties of dispersing billiards. Russ. Math. Surv. 25, 137–189 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sinai Ya.G., Chernov N.: Ergodic properties of some systems of two-dimensional discs and three-dimensional spheres. Russ. Math. Surv. 42, 181–207 (1987)

    Article  MathSciNet  Google Scholar 

  26. Young L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. 147, 585–650 (1998)

    Article  MATH  Google Scholar 

  27. Zhang H.-K.: Current in periodic Lorentz gases with twists. Commun. Math. Phys. 306(3), 747–776 (2011)

    Article  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Fairfield University, Fairfield, CT, 06824, USA

    Mark F. Demers

  2. Department of Mathematics and Statistics, UMass Amherst, MA, 01003, USA

    Hong-Kun Zhang

Authors
  1. Mark F. Demers
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hong-Kun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mark F. Demers.

Additional information

Communicated by G. Gallavotti

M. D. is partially supported by NSF Grant DMS-1101572.

H.-K. Z. is partially supported by NSF CAREER Grant DMS-1151762.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Demers, M.F., Zhang, HK. A Functional Analytic Approach to Perturbations of the Lorentz Gas. Commun. Math. Phys. 324, 767–830 (2013). https://doi.org/10.1007/s00220-013-1820-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-013-1820-0

Keywords

Search

Navigation