Skip to main content
SpringerLink
Log in

The localization transition of the two-dimensional Lorentz model

  • Regular Article
  • Published:
The European Physical Journal Special Topics Aims and scope Submit manuscript

Abstract.

We investigate the dynamics of a single tracer particle performing Brownian motion in a two-dimensional course of randomly distributed hard obstacles. At a certain critical obstacle density, the motion of the tracer becomes anomalous over many decades in time, which is rationalized in terms of an underlying percolation transition of the void space. In the vicinity of this critical density the dynamics follows the anomalous one up to a crossover time scale where the motion becomes either diffusive or localized. We analyze the scaling behavior of the time-dependent diffusion coefficient D(t) including corrections to scaling. Away from the critical density, D(t) exhibits universal hydrodynamic long-time tails both in the diffusive as well as in the localized phase.

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. M. Sahimi, Heterogeneous Materials, vol. 22 of Interdisciplinary Applied Mathematics (Springer, New York, 2003)

  2. G. Dagan, J. Fluid Mech. 145, 151 (1984)

    Article  MATH  ADS  Google Scholar 

  3. A. Meyer, J. Horbach, W. Kob, F. Kargl, H. Schober, Phys. Rev. Lett. 93, 027801 (2004)

    Article  ADS  Google Scholar 

  4. F. Kargl, A. Meyer, M.M. Koza, H. Schober, Phys. Rev. B 74, 014304 (2006)

    Article  ADS  Google Scholar 

  5. T. Voigtmann, J. Horbach, Europhys. Lett. 74, 459 (2006)

    Article  ADS  Google Scholar 

  6. A.J. Moreno, J. Colmenero, J. Chem. Phys. 125, 164507 (2006)

    Article  ADS  Google Scholar 

  7. A.J. Moreno, J. Colmenero, Phys. Rev. E 74, 021409 (2006)

    Article  ADS  Google Scholar 

  8. N. Kikuchi, J. Horbach, Europhys. Lett. 77, 26001 (2007)

    Article  ADS  Google Scholar 

  9. R.J. Ellis, Trends in Biochemical Sciences 26, 597 (2001)

    Article  Google Scholar 

  10. R.J. Ellis, A.P. Minton, Nature 425, 27 (2003)

    Article  ADS  Google Scholar 

  11. I.M. Tolić-Nørrelykke, E.-L. Munteanu, G. Thon, L. Oddershede, K. Berg-Sørensen, Phys. Rev. Lett. 93, 078102 (2004)

    Article  ADS  Google Scholar 

  12. I. Golding, E.C. Cox, Phys. Rev. Lett. 96, 098102 (2006)

    Article  ADS  Google Scholar 

  13. M.A. Deverall, E. Gindl, E.-K. Sinner, H. Besir, J. Ruehe, M.J. Saxton, C.A. Naumann, Biohpys. J. 88, 1875 (2005)

    Google Scholar 

  14. A. Kusumi, C. Nakada, K. Ritchie, K. Murase, K. Suzuki, H. Murakoshi, R.S. Kasai, J. Kondo, T. Fujiwara, Annu. Rev. Biophys. Biomol. Struct. 34, 351 (2005)

    Article  Google Scholar 

  15. M. Weiss, H. Hashimoto, T. Nilsson, Biophys. J. 84, 4043 (2003)

    Article  Google Scholar 

  16. M.R. Horton, F. Höfling, J.O. Rädler, T. Franosch, Soft Matter 6, 2648 (2010)

    Article  Google Scholar 

  17. H. van Beijeren, Rev. Mod. Phys. 54, 195 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  18. D. Stauffer, A. Aharony, Introduction to Percolation Theory, 2nd ed. (Taylor &amp Francis, London, 1994)

  19. F. Höfling, T. Franosch, E. Frey, Phys. Rev. Lett. 96, 165901 (2006)

    Article  ADS  Google Scholar 

  20. F. Höfling, T. Munk, E. Frey, T. Franosch, J. Chem. Phys. 128, 164517 (2008)

    Article  ADS  Google Scholar 

  21. F. Höfling, Ph.D. thesis, Ludwig-Maximilians-Universität München, 2006

  22. B.J. Sung, A. Yethiraj, Phys. Rev. Lett. 96, 228103 (2006)

    Article  ADS  Google Scholar 

  23. B.J. Sung, A. Yethiraj, J. Phys. Chem. B 112, 143 (2008)

    Article  Google Scholar 

  24. B.J. Sung, A. Yethiraj, J. Chem. Phys. 128, 054702 (2008)

    Article  ADS  Google Scholar 

  25. C. Bruin, Phys. Rev. Lett. 29, 1670 (1972)

    Article  ADS  Google Scholar 

  26. C. Bruin, Physica 72, 261 (1974)

    Article  ADS  Google Scholar 

  27. B.J. Alder, W.E. Alley, J. Stat. Phys. 19, 341 (1978)

    Article  ADS  Google Scholar 

  28. B.J. Alder, W.E. Alley, Physica A 121, 523 (1983)

    Article  ADS  Google Scholar 

  29. W.E. Alley, Ph.D. thesis, California Univ., Davis, 1979

  30. C.P. Lowe, A.J. Masters, Physica A 195, 149 (1993)

    Article  ADS  Google Scholar 

  31. A. Weijland, J.M.J. van Leeuwen, Physica (Amsterdam) 38, 35 (1968)

    Article  ADS  Google Scholar 

  32. M.H. Ernst, A. Weijland, Phys. Lett. A 34, 39 (1971)

    Article  ADS  Google Scholar 

  33. W. Götze, E. Leutheusser, S. Yip, Phys. Rev. A 23, 2634 (1981)

    Article  ADS  Google Scholar 

  34. W. Götze, E. Leutheusser, S. Yip, Phys. Rev. A 24, 1008 (1981)

    Article  ADS  Google Scholar 

  35. W. Götze, E. Leutheusser, S. Yip, Phys. Rev. A 25, 533 (1982)

    Article  ADS  Google Scholar 

  36. F. Höfling, T. Franosch, Phys. Rev. Lett. 98, 140601 (2007)

    Article  Google Scholar 

  37. V. Krakoviack, Phys. Rev. Lett. 94, 065703 (2005)

    Article  ADS  Google Scholar 

  38. V. Krakoviack, Phys. Rev. E 75, 031503 (2007)

    Article  ADS  Google Scholar 

  39. V. Krakoviack, Phys. Rev. E 79, 061501 (2009)

    Article  ADS  Google Scholar 

  40. J. Kurzidim, D. Coslovich, G. Kahl, Phys. Rev. Lett. 103, 138303 (2009)

    Article  ADS  Google Scholar 

  41. K. Kim, K. Miyazaki, S. Saito, Europhys. Lett. 88, 36002 (2009)

    Article  ADS  Google Scholar 

  42. J. Machta, S.M. Moore, Phys. Rev. A 32, 3164 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  43. D. ben-Avraham, S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems (Cambridge University Press, Cambridge, 2000)

  44. M.P.M. den Nijs, J. Phys. A 12, 1857 (1979)

    Article  ADS  Google Scholar 

  45. B. Nienhuis, Phys. Rev. Lett. 49, 1062 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  46. P. Grassberger, Physica A 262, 251 (1999)

    Article  MathSciNet  Google Scholar 

  47. A. Scala, T. Voigtmann, C. De Michele, J. Chem. Phys. 126, 134109 (2007)

    Article  ADS  Google Scholar 

  48. J. Quintanilla, S. Torquato, R.M. Ziff, J. Phys. A 33, L399 (2000)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  49. J.A. Quintanilla, R.M. Ziff, Phys. Rev. E 76, 051115 (2007)

    Article  ADS  Google Scholar 

  50. P.H. Colberg, F. Höfling, Accelerating glassy dynamics on graphics processing units (2009) [arXiv:0912.3824] [physics.comp-ph]

  51. T. Franosch, F. Höfling, T. Bauer, E. Frey, Chem. Phys. 375, 540 (2010)

    Article  ADS  Google Scholar 

  52. A. Kammerer, F. Höfling, T. Franosch, Europhys. Lett. 84, 66002 (2008)

    Article  ADS  Google Scholar 

  53. L. Santen, W. Krauth, Nature 405, 550 (2000)

    Article  ADS  Google Scholar 

  54. M. Bayer, J.M. Brader, F. Ebert, M. Fuchs, E. Lange, G. Maret, R. Schilling, M. Sperl, J.P. Wittmer, Phys. Rev. E 76, 011508 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  55. W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, 1968)

  56. W. Götze, L. Sjögren, J. Math. Anal. Appl. 195, 230 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  57. T. Franosch, W. Götze, J. Phys. Chem. B 103, 4011 (1999)

    Article  Google Scholar 

  58. T. Franosch, T. Voigtmann, J. Stat. Phys. 109, 237 (2002)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Arnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS), Fakultät für Physik, Ludwig-Maximilians-Universität München, Theresienstraße 37, 80333, München, Germany

    T. Bauer, T. Munk, E. Frey & T. Franosch

  2. Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford, OX1 3NP, UK

    F. Höfling

  3. Max-Planck-Institut für Metallforschung, Heisenbergstraße 3, 70569, Stuttgart, Germany

    F. Höfling

  4. Institut für Theoretische Physik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, 91058, Erlangen, Germany

    T. Franosch

  5. Institut für Theoretische und Angewandte Physik, Universität Stuttgart, Pfaffenwaldring 57, 70569, Stuttgart, Germany

    F. Höfling

Authors
  1. T. Bauer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Höfling
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. T. Munk
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. E. Frey
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. T. Franosch
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bauer, T., Höfling, F., Munk, T. et al. The localization transition of the two-dimensional Lorentz model. Eur. Phys. J. Spec. Top. 189, 103–118 (2010). https://doi.org/10.1140/epjst/e2010-01313-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjst/e2010-01313-1

Keywords

Search

Navigation