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The European Physical Journal E

, Volume 24, Issue 4, pp 367–377 | Cite as

Brownian particles in random and quasicrystalline potentials: How they approach the equilibrium

  • M. SchmiedebergEmail author
  • J. Roth
  • H. Stark
Regular Article

Abstract.

We study the Brownian motion of an ensemble of single colloidal particles in a random square and a quasicrystalline potential when they start from non-equlibrium. For both potentials, Brownian dynamics simulations reveal a widespread subdiffusive regime before the diffusive long-time limit is reached in thermal equilibrium. We develop a random trap model based on a distribution for the depths of trapping sites that reproduces the results of the simulations in detail. Especially, it gives analytic formulas for the long-time diffusion constant and the relaxation time into the diffusive regime. Aside from detailed differences, our work demonstrates that quasicrystalline potentials can be used to mimic aspects of random potentials.

PACS.

82.70.Dd Colloids 05.40.Jc Brownian motion 61.44.Br Quasicrystals 

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References

  1. 1.
    A. Einstein, Ann. Phys. (Leipzig) 17, 549 (1905).ADSGoogle Scholar
  2. 2.
    E. Frey, K. Kroy, Ann. Phys. (Leipzig) 14, 20 (2005).zbMATHCrossRefADSMathSciNetGoogle Scholar
  3. 3.
    A.V. Indrani, S. Ramaswamy, Phys. Rev. Lett. 73, 360 (1994).CrossRefADSGoogle Scholar
  4. 4.
    F.H. Stillinger, Science 267, 1935 (1995).CrossRefADSGoogle Scholar
  5. 5.
    C.A. Angell, Science 267, 1924 (1995)CrossRefADSGoogle Scholar
  6. 6.
    E.R. Weeks, J.C. Crocker, A.C. Levitt, A. Schofield, D.A. Weitz, Science 287, 627 (2000)CrossRefADSGoogle Scholar
  7. 7.
    D.G. Levitt, Phys. Rev. A 8, 3053 (1973).CrossRefADSGoogle Scholar
  8. 8.
    Q.-H. Wei, C. Bechinger, P. Leiderer, Science 287, 625 (2000).CrossRefADSGoogle Scholar
  9. 9.
    C. Lutz, M. Kollmann, C. Bechinger, Phys. Rev. Lett. 93, 026001 (2004).CrossRefADSGoogle Scholar
  10. 10.
    M. Kollmann, Phys. Rev. Lett. 90, 180602 (2003).CrossRefADSGoogle Scholar
  11. 11.
    F. Höfling, T. Franosch, E. Frey, Phys. Rev. Lett. 96, 165901 (2006).CrossRefADSGoogle Scholar
  12. 12.
    Y. Kafri, D.K. Lubensky, D.R. Nelson, Biophys. J. 86, 3373 (2004).CrossRefADSGoogle Scholar
  13. 13.
    M. Weiss, M. Elsner, F. Kartberg, T. Nilsson, Biophys. J. 87, 3518 (2004).CrossRefADSGoogle Scholar
  14. 14.
    F. Amblard, Phys. Rev. Lett. 77, 4470 (1996).CrossRefADSGoogle Scholar
  15. 15.
    P. Bursac, G. Lenormand, B. Fabry, M. Oliver, D.A. Weitz, V. Viasnoff, J.P. Butler, J.J. Fredberg, Nat. Mater. 4, 557 (2005).CrossRefADSGoogle Scholar
  16. 16.
    P. Smith, I.E.G. Morrison, K.M. Wilson, N. Fernandez, R.J. Cherry, Biophys. J. 76, 3331 (1999).ADSGoogle Scholar
  17. 17.
    P. Schwille, U. Haupts, S. Maiti, W.W. Webb, Biophys. J. 77, 2251 (1999).ADSGoogle Scholar
  18. 18.
    I. Golding, E.C. Cox, Phys. Rev. Lett. 96, 098102 (2006).CrossRefADSGoogle Scholar
  19. 19.
    T.J. Feder, I. Brust-Mascher, J.P. Slattery, B. Baird, W.W. Webb, Biophys. J. 70, 2767 (1996).CrossRefADSGoogle Scholar
  20. 20.
    J. Honkonen, Y.M. Pis'mak, J. Phys. A 22, L899 (1989).Google Scholar
  21. 21.
    A.H. Romero, J.M. Sancho, Phys. Rev. E 58, 2833 (1998).CrossRefADSGoogle Scholar
  22. 22.
    J.M. Sancho, A.M. Lacasta, K. Lindenberg, I.M. Sokolov, A.H. Romero, Phys. Rev. Lett. 92, 250601 (2004).CrossRefADSGoogle Scholar
  23. 23.
    A.M. Lacasta, J.M. Sancho, A.H. Romero, I.M. Sokolov, K. Lindenberg, Phys. Rev. E 70, 051104 (2004). CrossRefADSGoogle Scholar
  24. 24.
    J.W. Haus, K.W. Kehr, J.W. Lyklema, Phys. Rev. B 25, 2905 (1982).CrossRefADSGoogle Scholar
  25. 25.
    J.W. Haus, K.W. Kehr, Phys. Rep. 150, 263 (1987).CrossRefADSGoogle Scholar
  26. 26.
    Derrida, J. Stat. Phys. 31, 433 (1983).CrossRefADSMathSciNetGoogle Scholar
  27. 27.
    D. Shechtman, I. Blech, D. Gratias, J.W. Cahn, Phys. Rev. Lett. 53, 1951 (1984).CrossRefADSGoogle Scholar
  28. 28.
    D.A. Rabson, N.D. Mermin, D.S. Rokhsar, D.C. Wright, Rev. Mod. Phys. 63, 699 (1991).CrossRefADSMathSciNetGoogle Scholar
  29. 29.
    B. Passaro, C. Sire, V.G. Benza, Phys. Rev. B 46, 13751 (1992).CrossRefADSGoogle Scholar
  30. 30.
    L. Guidoni, B. Dépret, A. di Stefano, P. Verkerk, Phys. Rev. A 60, R4233 (1999).Google Scholar
  31. 31.
    R. McGrath, J. Ledieu, E.J. Cox, R.D. Diehl, J. Phys.: Condens. Matter 14, R119 (2002).Google Scholar
  32. 32.
    M. Shimoda, T.J. Sato, A.P. Tsai, J.Q. Guo, Phys. Rev. B 62, 11288 (2000).CrossRefADSGoogle Scholar
  33. 33.
    K.J. Franke, H.R. Sharma, W. Theis, P. Gille, Ph. Ebert, K.H. Rieder, Phys. Rev. Lett. 89, 156104 (2002).CrossRefADSGoogle Scholar
  34. 34.
    V. Fournée, T.C. Cai, A.R. Ross, T.A. Lograsso, J.W. Evans, P.A. Thiel, Phys. Rev. B 67, 033406 (2003).CrossRefADSGoogle Scholar
  35. 35.
    J. Ledieu, J.T. Hoeft, D.E. Reid, J.A. Smerdon, R.D. Diehl, T.A. Lograsso, A.R. Ross, R. McGrath, Phys. Rev. Lett. 92, 135507 (2004).CrossRefADSGoogle Scholar
  36. 36.
    R.A. Tasca, N. Ferralis, R.D. Diehl M.W. Cole, J. Phys.: Condens. Matter 16, S2911 (2004).Google Scholar
  37. 37.
    S. Curtarolo, W. Setyawan, N. Ferralis, R.D. Diehl, M.W. Cole, Phys. Rev. Lett. 95, 136104 (2005).CrossRefADSGoogle Scholar
  38. 38.
    A. Ashkin, Phys. Rev. Lett. 24, 156 (1970)CrossRefADSGoogle Scholar
  39. 39.
    M.M. Burns, J.M. Fournier, J.A. Golovchenko, Science 249, 749 (1990).CrossRefADSGoogle Scholar
  40. 40.
    A. Chowdhury, B.J. Ackerson, N.A. Clark, Phys. Rev. Lett. 55, 833 (1985).CrossRefADSGoogle Scholar
  41. 41.
    Q.-H. Wei, C. Bechinger, D. Rudhardt, P. Leiderer, Phys. Rev. Lett. 81, 2606 (1998)CrossRefADSGoogle Scholar
  42. 42.
    J. Chakrabarti, H.R. Krishnamurthy, A.K. Sood, Phys. Rev. Lett. 73, 2923 (1994)CrossRefADSGoogle Scholar
  43. 43.
    M. Schmiedeberg, J. Roth, H. Stark, Phys. Rev. Lett. 97, 158304 (2006).CrossRefADSGoogle Scholar
  44. 44.
    For example it takes approximatly 90s CPU time per million steps per particle on one unit of an AMD Athlon64 X2 Dual-Core 4600+ (2.4 GHz) processor.Google Scholar
  45. 45.
    S.P. Gorkhali, J. Qi, G.P. Crawford, J. Opt. Soc. Am. B 23, 149 (2005).CrossRefADSGoogle Scholar
  46. 46.
    A.A. Chernikov, R.Z. Sagdeev, D.A. Usikov, G.M. Zaslavsky, Phys. Lett. A 125, 101 (1987).CrossRefADSMathSciNetGoogle Scholar
  47. 47.
    D. Levine, T.C. Lubensky, S. Ostlund, S. Ramaswamym, P.J. Steinhardt, J. Toner, Phys. Rev. Lett. 54, 1520 (1985)CrossRefADSGoogle Scholar
  48. 48.
    P. Hänggi, P. Talkner, M. Borkovec, Rev. Mod. Phys. 62, 251 (1990).CrossRefADSGoogle Scholar
  49. 49.
    R. Collins, J. Phys. C 1, 1461 (1968)CrossRefADSGoogle Scholar

Copyright information

© EDP Sciences, Società Italiana di Fisica and Springer-Verlag 2008

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikTechnische Universität BerlinBerlinGermany
  2. 2.Institut für Theoretische und Angewandte PhysikUniversität StuttgartStuttgartGermany

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