Active Brownian particles moving in a random Lorentz gas

  • Maria ZeitzEmail author
  • Katrin Wolff
  • Holger Stark
Open Access
Regular Article


Biological microswimmers often inhabit a porous or crowded environment such as soil. In order to understand how such a complex environment influences their spreading, we numerically study non-interacting active Brownian particles (ABPs) in a two-dimensional random Lorentz gas. Close to the percolation transition in the Lorentz gas, they perform the same subdiffusive motion as ballistic and diffusive particles. However, due to their persistent motion they reach their long-time dynamics faster than passive particles and also show superdiffusive motion at intermediate times. While above the critical obstacle density \(\eta_{c}\) the ABPs are trapped, their long-time diffusion below \(\eta_{c}\) is strongly influenced by the propulsion speed v0. With increasing v0, ABPs are stuck at the obstacles for longer times. Thus, for large propulsion speed, the long-time diffusion constant decreases more strongly in a denser obstacle environment than for passive particles. This agrees with the behavior of an effective swimming velocity and persistence time, which we extract from the velocity autocorrelation function.

Graphical abstract


Soft Matter: Colloids and Nanoparticles 


  1. 1.
    S. Ramaswamy, Annu. Rev. Condens. Matter Phys. 1, 323 (2010)ADSCrossRefGoogle Scholar
  2. 2.
    P. Romanczuk, M. Bär, W. Ebeling, B. Lindner, L. Schimansky-Geier, Eur. Phys. J. ST 202, 1 (2012)CrossRefGoogle Scholar
  3. 3.
    M.C. Marchetti, J.F. Joanny, S. Ramaswamy, T.B. Liverpool, J. Prost, M. Rao, R.A. Simha, Rev. Mod. Phys. 85, 1143 (2013)ADSCrossRefGoogle Scholar
  4. 4.
    A. Zöttl, H. Stark, J. Phys.: Condens. Matter 28, 253001 (2016)ADSGoogle Scholar
  5. 5.
    M. Polin, I. Tuval, K. Drescher, J.P. Gollub, R.E. Goldstein, Science 325, 487 (2009)ADSCrossRefGoogle Scholar
  6. 6.
    D. Alizadehrad, T. Krüger, M. Engstler, H. Stark, PLoS Comput. Biol. 11, e1003967 (2015)ADSCrossRefGoogle Scholar
  7. 7.
    M. Schmitt, H. Stark, Europhys. Lett. 101, 44008 (2013)ADSCrossRefGoogle Scholar
  8. 8.
    T.C. Adhyapak, H. Stark, Phys. Rev. E 92, 052701 (2015)ADSCrossRefGoogle Scholar
  9. 9.
    M. Schmitt, H. Stark, Eur. Phys. J. E 39, 80 (2016)ADSCrossRefGoogle Scholar
  10. 10.
    C.C. Maass, C. Krüger, S. Herminghaus, C. Bahr, Annu. Rev. Condens. Matter Phys. 7, 171 (2016)ADSCrossRefGoogle Scholar
  11. 11.
    J.R. Howse, R.A.L. Jones, A.J. Ryan, T. Gough, R. Vafabakhsh, R. Golestanian, Phys. Rev. Lett. 99, 8 (2007)CrossRefGoogle Scholar
  12. 12.
    K. Drescher, J. Dunkel, L.H. Cisneros, S. Ganguly, R.E. Goldstein, Proc. Natl. Acad. Sci. U.S.A. 108, 10940 (2011)ADSCrossRefGoogle Scholar
  13. 13.
    A.P. Berke, L. Turner, H.C. Berg, E. Lauga, Phys. Rev. Lett. 101, 038102 (2008)ADSCrossRefGoogle Scholar
  14. 14.
    K. Schaar, A. Zöttl, H. Stark, Phys. Rev. Lett. 115, 038101 (2015)ADSCrossRefGoogle Scholar
  15. 15.
    D. Takagi, J. Palacci, A.B. Braunschweig, M.J. Shelley, J. Zhang, Soft Matter 10, 1784 (2014)ADSCrossRefGoogle Scholar
  16. 16.
    A. Kaiser, H.H. Wensink, H. Löwen, Phys. Rev. Lett. 108, 268307 (2012)ADSCrossRefGoogle Scholar
  17. 17.
    H.C. Berg, E. Coli in Motion, Biological and Medical Physics, Biomedical Engineering (Springer New York, New York, NY, 2004). -0.3ptGoogle Scholar
  18. 18.
    R.M. Ford, R.W. Harvey, Adv. Water Resour. 30, 1608 (2007)ADSCrossRefGoogle Scholar
  19. 19.
    M. Engstler, T. Pfohl, S. Herminghaus, M. Boshart, G. Wiegertjes, N. Heddergott, P. Overath, Cell 131, 505 (2007)CrossRefGoogle Scholar
  20. 20.
    S. Park, H. Hwang, S.W. Nam, F. Martinez, R.H. Austin, W.S. Ryu, PLoS One 3, 1 (2008)Google Scholar
  21. 21.
    S.H. Holm, J.P. Beech, M.P. Barrett, J.O. Tegenfeldt, Lab Chip 11, 1326 (2011)CrossRefGoogle Scholar
  22. 22.
    G. Volpe, I. Buttinoni, D. Vogt, H.-J. Kümmerer, C. Bechinger, Soft Matter 7, 8810 (2011)ADSCrossRefGoogle Scholar
  23. 23.
    N. Heddergott, T. Krüger, S.B. Babu, A. Wei, E. Stellamanns, S. Uppaluri, T. Pfohl, H. Stark, M. Engstler, PLoS Pathog. 8, 11 (2012)CrossRefGoogle Scholar
  24. 24.
    T. Majmudar, E.E. Keaveny, J. Zhang, M.J. Shelley, J. R. Soc. Interface 9, 1809 (2012)CrossRefGoogle Scholar
  25. 25.
    S. Johari, V. Nock, M.M. Alkaisi, W. Wang, Lab Chip 13, 1699 (2013)CrossRefGoogle Scholar
  26. 26.
    O. Chepizhko, F. Peruani, Phys. Rev. Lett. 111, 16 (2013)CrossRefGoogle Scholar
  27. 27.
    O. Chepizhko, E.G. Altmann, F. Peruani, Phys. Rev. Lett. 110, 23 (2013)CrossRefGoogle Scholar
  28. 28.
    C. Reichhardt, C.J. Olson Reichhardt, Phys. Rev. E 90, 1 (2014)CrossRefGoogle Scholar
  29. 29.
    W. Schirmacher, B. Fuchs, F. Höfling, T. Franosch, Phys. Rev. Lett. 115, 240602 (2015)ADSCrossRefGoogle Scholar
  30. 30.
    O. Chepizhko, F. Peruani, Eur. Phys. J. ST 224, 1287 (2015)CrossRefGoogle Scholar
  31. 31.
    M. Raatz, M. Hintsche, M. Bahrs, M. Theves, C. Beta, Eur. Phys. J. ST 224, 1185 (2015)CrossRefGoogle Scholar
  32. 32.
    J.L. Münch, D. Alizadehrad, S.B. Babu, H. Stark, Soft Matter 12, 7350 (2016)ADSCrossRefGoogle Scholar
  33. 33.
    M. Khatami, K. Wolff, O. Pohl, M.R. Ejtehadi, H. Stark, Sci. Rep. 6, 37670 (2016)ADSCrossRefGoogle Scholar
  34. 34.
    J. Tailleur, M.E. Cates, EPL 86, 60002 (2009)ADSCrossRefGoogle Scholar
  35. 35.
    M.B. Wan, C.J. Olson Reichhardt, Z. Nussinov, C. Reichhardt, Phys. Rev. Lett. 101, 018102 (2008)ADSCrossRefGoogle Scholar
  36. 36.
    G. Volpe, S. Gigan, G. Volpe, Am. J. Phys. 82, 659 (2014)ADSCrossRefGoogle Scholar
  37. 37.
    P. Galajda, J. Keymer, P. Chaikin, R. Austin, J. Bacteriol. 189, 8704 (2007)CrossRefGoogle Scholar
  38. 38.
    C.J.O. Reichhardt, C. Reichhardt, arXiv:1604.01072 [cond-mat.soft] (2016). -0.3pt
  39. 39.
    C. Sándor, A. Libál, C. Reichhardt, C.J.O. Reichhardt, arXiv:1608.05323 [cond-mat.soft] (2016). -0.3pt
  40. 40.
    G. Volpe, G. Volpe, S. Gigan, Sci. Rep. 4, 3936 (2014)ADSCrossRefGoogle Scholar
  41. 41.
    A. Pototsky, A.M. Hahn, H. Stark, Phys. Rev. E 87, 042124 (2013)ADSCrossRefGoogle Scholar
  42. 42.
    C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, G. Volpe, G. Volpe, Rev. Mod. Phys. 88, 045006 (2016)ADSCrossRefGoogle Scholar
  43. 43.
    A. Weijland, J. Van Leeuwen, Physica 38, 35 (1968)ADSCrossRefGoogle Scholar
  44. 44.
    F. Hofling, T. Munk, E. Frey, T. Franosch, J. Chem. Phys. 128, 164517 (2008)ADSCrossRefGoogle Scholar
  45. 45.
    S. Schöbl, S. Sturm, W. Janke, K. Kroy, Phys. Rev. Lett. 113, 238302 (2014)ADSCrossRefGoogle Scholar
  46. 46.
    S. Mertens, C. Moore, Phys. Rev. E 86, 061109 (2012)ADSCrossRefGoogle Scholar
  47. 47.
    M. Spanner, F. Höfling, G.E. Schröder-Turk, K. Mecke, T. Franosch, J. Phys.: Condens. Matter 23, 234120 (2011)ADSGoogle Scholar
  48. 48.
    T. Bauer, F. Höfling, T. Munk, E. Frey, T. Franosch, Eur. Phys. J. ST 189, 103 (2010)CrossRefGoogle Scholar
  49. 49.
    M. Spanner, F. Höfling, S.C. Kapfer, K.R. Mecke, G.E. Schröder-Turk, T. Franosch, Phys. Rev. Lett. 116, 060601 (2016)ADSCrossRefGoogle Scholar
  50. 50.
    S.K. Schnyder, M. Spanner, F. Höfling, T. Franosch, J. Horbach, Soft Matter 11, 701 (2015)ADSCrossRefGoogle Scholar
  51. 51.
    C. Reichhardt, C.J. Olson Reichhardt, Soft Matter 10, 7502 (2014)ADSCrossRefGoogle Scholar
  52. 52.
    D. Stauffer, A. Aharony, Introduction to Percolation Theory (Taylor & Francis, 1994). -0.3ptGoogle Scholar
  53. 53.
    S. Torquato, Random Heterogeneous Materials, Vol. 16 of Interdisciplinary Applied Mathematics (Springer New York, New York, NY, 2002). -0.3ptGoogle Scholar
  54. 54.
    M.T. Downton, H. Stark, J. Phys.: Condens. Matter 21, 204101 (2009)ADSGoogle Scholar
  55. 55.
    B. ten Hagen, S. van Teeffelen, H. Löwen, J. Phys. Condens. Matter 23, 194119 (2011)ADSCrossRefGoogle Scholar
  56. 56.
    R. Kapral, J. Chem. Phys. 138, 020901 (2013)ADSCrossRefGoogle Scholar
  57. 57.
    J.K.G. Dhont, An Introduction to Dynamics of Colloids (Elsevier, 1996). -0.3ptGoogle Scholar
  58. 58.
    R. Kubo, M. Toda, N. Hashitsume, Statistical Physics II: Nonequilibrium Statistical Mechanics (Springer, Berlin, Heidelberg, 1991). -0.3ptGoogle Scholar
  59. 59.
    C.M. Lo, H.B. Wang, M. Dembo, Y.L. Wang, Biophys. J. 79, 144 (2000)CrossRefGoogle Scholar
  60. 60.
    S.V. Plotnikov, C.M. Waterman, Curr. Opin. Cell Biol. 25, 619 (2013)CrossRefGoogle Scholar
  61. 61.
    T. Speck, A.M. Menzel, J. Bialké, H. Löwen, J. Chem. Phys. 142, 224109 (2015)ADSCrossRefGoogle Scholar
  62. 62.
    M.E. Cates, J. Tailleur, Annu. Rev. Condens. Matter Phys. 6, 219 (2015)ADSCrossRefGoogle Scholar
  63. 63.
    S.C. Takatori, J.F. Brady, Phys. Rev. E 91, 032117 (2015)ADSCrossRefGoogle Scholar
  64. 64.
    J. Blaschke, M. Maurer, K. Menon, A. Zöttl, H. Stark, Soft Matter 12, 9821 (2016)ADSCrossRefGoogle Scholar

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© The Author(s) 2017

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikTechnische Universität BerlinBerlinGermany

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