The European Physical Journal Special Topics

, Volume 187, Issue 1, pp 49–62 | Cite as

The rich phenomenology of Brownian particles in nonlinear potential landscapes

  • J.M. SanchoEmail author
  • A.M. Lacasta


Non-interacting Brownian particles obey Langevin equations fulfilling a fluctuation–dissipation relation between friction and thermal noise. Under a linear potential (constant force) Einstein found a relation between diffusion and transport through mobility. In nonlinear potentials this prediction is only satisfied within the limits of very small and large constant external forces. Moreover, other more interesting behaviors do appear, such as: dispersionless transport, sorting, giant diffusion, subdiffusion, superdiffusion, subtransport, etc. All these phenomena depend on the characteristics of the nonlinear potential landscape: periodic or random, the symmetries and boundary conditions. Moreover, the presence of transport is the keystone of most of this phenomenology. In this review, we present numerical simulations illustrating these facts and theoretical analysis when possible.


European Physical Journal Special Topic Brownian Particle Periodic Potential Critical Force Random Potential 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    A. Einstein, Ann. Phys. 17, 549 (1905)CrossRefGoogle Scholar
  2. 2.
    V. Blickle, T. Speck, C. Lutz, U. Seifert, C. Bechinger, Phys. Rev. Lett. 98, 210601 (2007)CrossRefMathSciNetADSGoogle Scholar
  3. 3.
    M.P. MacDonald, G.C. Spalding, K. Dholakia, Nature (London) 426, 421 (2003)CrossRefADSGoogle Scholar
  4. 4.
    D.G. Grier, Nature (London) 424, 810 (2003)CrossRefADSGoogle Scholar
  5. 5.
    P.T. Korda, M.B. Taylor, D.G. Grier, Phys. Rev. Lett. 89, 128301 (2002)CrossRefADSGoogle Scholar
  6. 6.
    L.R. Huang, E.C. Cox, R.H. Austin, J.C. Sturm, Science 304, 987 (2004)CrossRefADSGoogle Scholar
  7. 7.
    K.J. Morton, K. Loutherback, D.W. Inglis, O.K. Tsui, J.C. Sturm, S.Y. Chou, R.H. Austin, Proc. Natl. Acad. Sci. USA 105, 7434 (2008)CrossRefADSGoogle Scholar
  8. 8.
    K. Lindenberg, J.M. Sancho, A.M. Lacasta, I.M. Sokolov, Phys. Rev. Lett. 98, 020602 (2007)CrossRefADSGoogle Scholar
  9. 9.
    K. Lindenberg, A.M. Lacasta, J.M. Sancho, A.H. Romero, New J. Phys. 7, 29 (2005)CrossRefGoogle Scholar
  10. 10.
    P. Colet, M. San Miguel, J.M. Sancho, Phys. Rev. A 39, 149 (1989)CrossRefADSGoogle Scholar
  11. 11.
    B. Lindner, M. Kostur, L. Schimansky-Geier, Fluct. Noise Lett. 1, R25 (2001)CrossRefGoogle Scholar
  12. 12.
    P. Reimann, C. Van den Broeck, H. Linke, P. Hanggi, J.M. Rubi, A. Pérez-Madrid, Phys. Rev. Lett. 87, 010602 (2001)CrossRefADSGoogle Scholar
  13. 13.
    S.H. Lee, D.G. Grier, Phys. Rev. Lett. 96, 190601 (2006)CrossRefADSGoogle Scholar
  14. 14.
    A.M. Lacasta, J.M. Sancho, A.H. Romero, K. Lindenberg, Phys. Rev. Lett. 94, 160601 (2005)CrossRefADSGoogle Scholar
  15. 15.
    J.M. Sancho, M. Khoury, K. Lindenberg, A.M. Lacasta, J. Phys. Condensed Matter 17, S4151 (2005)CrossRefADSGoogle Scholar
  16. 16.
    A.M. Lacasta, M. Khoury, J.M. Sancho, K. Lindenberg, Mod. Phys. Lett. B 20, 1427 (2006)zbMATHCrossRefADSGoogle Scholar
  17. 17.
    J.P. Gleeson, J.M. Sancho, A.M. Lacasta, K. Lindenberg, Phys. Rev. E 73, 041102 (2006)CrossRefMathSciNetADSGoogle Scholar
  18. 18.
    J.M. Sancho, A.M. Lacasta, K. Lindenberg, I.M. Sokolov, A.H. Romero, Phys. Rev. Lett. 92, 250601 (2004)CrossRefADSGoogle Scholar
  19. 19.
    A.M. Lacasta, J.M. Sancho, A.H. Romero, I.M. Sokolov, K. Lindenberg, Phys. Rev. E 70, 051104 (2004)CrossRefADSGoogle Scholar
  20. 20.
    A.M. Lacasta, J.M. Sancho (2010) (preprint)Google Scholar
  21. 21.
    A.H. Romero, A.M. Lacasta, J.M. Sancho, Phys. Rev. E 69, 051105 (2004)CrossRefADSGoogle Scholar
  22. 22.
    E. Heinsalu, M. Patriarca, F. Marchesoni, Phys. Rev. E 77, 021129 (2008)CrossRefADSGoogle Scholar
  23. 23.
    D. Hennig, S. Martens, S. Fugmann, Phys. Rev. E 78, 011104 (2008)CrossRefADSGoogle Scholar
  24. 24.
    M. Tiwari, S. Gonçalves, V.M. Kenkre, Eur. Phys. J. B 62, 459 (2008)CrossRefADSGoogle Scholar
  25. 25.
    S. Gonçalves, C. Fusco, A.R. Bishop, V.M. Kenkre, Phys. Rev. E 72, 195418 (2005)ADSGoogle Scholar
  26. 26.
    S. von Gehlen, M. Evstigneev, P. Reimann, Phys. Rev. E 77, 031136 (2008)CrossRefADSGoogle Scholar
  27. 27.
    A. Taloni, F. Marchesoni, Phys. Rev. Lett. 96, 020601 (2006)CrossRefADSGoogle Scholar
  28. 28.
    H. Risken, The Fokker–Planck Equation (Springer-Verlag, New York, 1989)Google Scholar
  29. 29.
    C. Costantini, F. Marchesoni, Europhys. Lett. 48, 491 (1999)CrossRefADSGoogle Scholar
  30. 30.
    J.G. Giddings, Unified Separation Science (John Wiley, New York, 1991)Google Scholar
  31. 31.
    C. Reichhardt, C.J. Olson, Phys. Rev. Lett. 89, 078301 (2002)CrossRefADSGoogle Scholar
  32. 32.
    P. Reimann, R. Eichhorn, Phys. Rev. Lett. 101, 180601 (2008)CrossRefADSGoogle Scholar
  33. 33.
    J. García-Ojalvo, J.M. Sancho, Noise in Spatially Extended System (Springer, New York, 1999)Google Scholar
  34. 34.
    M. Khoury, J.O. Gleeson, J.M. Sancho, A.M. Lacasta, K. Lindenberg, Phys. Rev. E 80, 021123 (2009)CrossRefADSGoogle Scholar
  35. 35.
    A.H. Romero, J.M. Sancho, Phys. Rev. E 58, 2833 (1998)CrossRefADSGoogle Scholar
  36. 36.
    J.-P. Bouchaud, A. Georges, Phys. Rep. 195, 127 (1990)CrossRefMathSciNetADSGoogle Scholar
  37. 37.
    J.M. SanchoA.M. Lacasta, K. Lindenberg, I.M. Sokolov, A.H. Romero, Phys. Rev. Lett. 94, 188902 (2005)CrossRefADSGoogle Scholar
  38. 38.
    M. Khoury, A.M. Lacasta, J.M. Sancho, K. Lindenberg (2010) (preprint)Google Scholar
  39. 39.
    A.V. Chechkin, R. Gorenflo, I.M. Sokolov, Phys. Rev. E 66, 046129 (2002)CrossRefADSGoogle Scholar
  40. 40.
    P. Tierno, F. Sagues, T.H. Johansen, T.M. Fisher, Phys. Chem. Chem. Phys. 11, 9615 (2009)CrossRefGoogle Scholar
  41. 41.
    M. Khoury, A.M. Lacasta, J.M. Sancho, A.H. Romero, K. Lindenberg, Phys. Rev. B 78, 155433 (2008)CrossRefADSGoogle Scholar

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© EDP Sciences and Springer 2010

Authors and Affiliations

  1. 1.Departament d’Estructura i Constituents de la MatèriaUniversitat de BarcelonaBarcelonaSpain
  2. 2.Departament de Física AplicadaUniversitat Politèctica de CatalunyaBarcelonaSpain

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