Langevin analysis for time-nonlocal Brownian motion with algebraic memories and delay interactions

  • Matthew Chase
  • Tom J. McKetterick
  • Luca Giuggioli
  • V. M. Kenkre
Regular Article


Starting from a Langevin equation with memory describing the attraction of a particle to a center, we investigate its transport and response properties corresponding to two special forms of the memory: one is algebraic, i.e., power-law, and the other involves a delay. We examine the properties of the Green function of the Langevin equation and encounter Mittag-Leffler and Lambert W-functions well-known in the literature. In the presence of white noise, we study two experimental situations, one involving the motional narrowing of spectral lines and the other the steady-state size of the particle under consideration. By comparing the results to counterparts for a simple exponential memory, we uncover instructive similarities and differences. Perhaps surprisingly, we find that the Balescu-Swenson theorem that states that non-Markoffian equations do not add anything new to the description of steady-state or equilibrium observables is violated for our system in that the saturation size of the particle in the steady-state depends on the memory function utilized. A natural generalization of the Smoluchowski equation for the time-local case is examined and found to satisfy the Balescu-Swenson theorem and describe accurately the first moment but not the second and higher moments. We also calculate two-time correlation functions for all three cases of the memory, and show how they differ from (tend to) their Markoffian counterparts at small (large) values of the difference between the two times.


Statistical and Nonlinear Physics 


  1. 1.
    H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications, 2nd edn. (Springer-Verlag, Berlin, 1989)Google Scholar
  2. 2.
    L.E. Reichl, A Modern Course in Statistical Physics (WileyVCH Verlag, Weinheim, 2009)Google Scholar
  3. 3.
    A.A. Budini, M.O. Cáceres, J. Phys. A 37, 5959 (2004)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    A.A. Budini, M.O. Cáceres, Physica A 356, 31 (2005)ADSCrossRefGoogle Scholar
  5. 5.
    A.D. Drozdov, Physica A 376, 237 (2007)ADSCrossRefGoogle Scholar
  6. 6.
    J.E. Fiscina, M.O. Cáceres, F. Mücklich, J. Phys.: Condens. Matter 17, S1237 (2005)ADSGoogle Scholar
  7. 7.
    A.O. Bolivar, Physica A 390, 3095 (2011)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    A.K. Das, S. Panda, J.R.L. Santos, Int. J. Mod. Phys. A 30, 1550028 (2015)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    P. Hänggi, P. Talkner, Phys. Lett. A 68, 9 (1978)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    P. Hänggi, H. Thomas, H. Grabert, P. Talkner, J. Stat. Phys. 18, 155 (1978)ADSCrossRefGoogle Scholar
  12. 12.
    P. Hänggi, in Stochastic Processes Applied to Physics, edited by L. Pesquera, M. Rodriguez (World Scientific, Philadelphia, 1985), p. 69.Google Scholar
  13. 13.
    P. Hänggi, in Noise in Nonlinear Dynamical Systems, edited by F. Moss, P.V.E. McClintock (Cambridge University Press, Cambridge, 1989), p. 307Google Scholar
  14. 14.
    L. Giuggioli, V.M. Kenkre, Movement Ecology 2, 20 (2014)CrossRefGoogle Scholar
  15. 15.
    V.M. Kenkre, arXiv:0708.0034 (2007)
  16. 16.
    T.J. McKetterick, L. Giuggioli, Phys. Rev. E 90, 042135 (2014)ADSCrossRefGoogle Scholar
  17. 17.
    L. Giuggioli, T.J. McKetterick, M. Holderied, PLoS Comput. Biol. 11, e1004089 (2015)ADSCrossRefGoogle Scholar
  18. 18.
    K. Spendier, S. Sugaya, V.M. Kenkre, Phys. Rev. E 88, 062142 (2013)ADSCrossRefGoogle Scholar
  19. 19.
    V.M. Kenkre, S. Sugaya, Bull. Math. Bio. 76, 3016 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    T.L. Yates et al., Bioscience 52, 989 (2002)CrossRefGoogle Scholar
  21. 21.
    C. Fuchs, Inference for Diffusion Processes: with Applications in life Sciences (Springer, Berlin, 2013)Google Scholar
  22. 22.
    P.Z. Marmarelis, V.Z. Marmarelis, Analysis of Physiological Systems: the White-noise Approach (Plenum Press, New York, 1978)Google Scholar
  23. 23.
    D.J. Wilkinson, Stochastic Modelling for Systems Biology, 2nd edn. (CRC Press, Boca Raton, 2012)Google Scholar
  24. 24.
    A. Okubo S.A. Levin, Diffusion and Ecological Problems: Modern Perspectives, 2nd edn. (Springer, New York, 2002)Google Scholar
  25. 25.
    D. Boyer, C. Solis-Salas, Phys. Rev. Lett. 112, 240601 (2014)ADSCrossRefGoogle Scholar
  26. 26.
    S. Trimper, G.M. Schutz, Phys. Rev. E 70, 045101(R) (2004)ADSCrossRefGoogle Scholar
  27. 27.
    E.A. Novikov, Sov. Phys. J. Exp. Theor. Phys. 20, 1290 (1965)Google Scholar
  28. 28.
    N.G. van Kampen, Stochastic Process in Physics and Chemistry (Elsevier, Amsterdam, 2007)Google Scholar
  29. 29.
    A.S. Adelman, J. Chem. Phys. 64, 724 (1976)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    R.F. Fox, J. Stat. Phys. 18, 2331 (1977)ADSGoogle Scholar
  31. 31.
    P. Hänggi, Z. Phys. B 31, 407 (1978)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    M. San Miguel, J.M. Sancho, J. Stat. Phys. 22, 605 (1980)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications (Springer-Verlag, Berlin, 2014)Google Scholar
  34. 34.
    M. Abramowitz, I.E. Stegun Handbook of Mathematical Functions (Dover Publications, Toronto, 1970)Google Scholar
  35. 35.
    R.E. Bellman, K.L. Cooke, Differential-difference Equations (Academic Press Inc., New York, 1963)Google Scholar
  36. 36.
    F.M. Asl, A.G. Ulsoy, J. Dyn. Syst. Meas. Contr. 125, 215 (2003)CrossRefGoogle Scholar
  37. 37.
    A.A. Budini, M.O. Cáceres, Phys. Rev. E 70, 046104 (2004)ADSCrossRefGoogle Scholar
  38. 38.
    U. Küchler, B. Mensch, Stoc. Stoc. Rep. 40, 23 (1992)CrossRefGoogle Scholar
  39. 39.
    L. Euler, Acta Acad. Sci. Petropol. 2, 29 (1783)Google Scholar
  40. 40.
    R.M. Corless et al., Adv. Comput. Math. 5, 329 (1996)MathSciNetCrossRefGoogle Scholar
  41. 41.
    V.M. Kenkre, Phys. Lett. A 47, 119 (1974)ADSCrossRefGoogle Scholar
  42. 42.
    V.M. Kenkre, in Exciton Dynamics in Molecular Crystals and Aggregates, Springer Tracts in Modern Physics, edited by G. Hoehler (Springer, Berlin, 1982), Vol. 94, p. 1Google Scholar
  43. 43.
    V.M. Kenkre, in Proceedings of the NATO Advance Study Institute on Energy Transfer, Erice, Italy, June 15–30, 1983, edited by B. DiBartolo (Plenum, New York, 1984), p. 205Google Scholar
  44. 44.
    R. Kubo, in Fluctuation, Relaxation and Resonance in Magnetic Systems Scottish Universities’ Summer School, edited by D. ter Haar (Oliver and Boyd, Edinburgh, 1961), p. 23Google Scholar
  45. 45.
    C.P. Slichter, Principles of Magnetic Resonance, 2nd edn. (Spring-Verlag, Berlin, 1978), see especially, p. 374Google Scholar
  46. 46.
    T. Springer, in Quasielastic Neutron Scattering for the Investigation of Diffusive Motions in Solids and Liquids, Springer Tracts in Modern Physics (Springer, Berlin, 1972), Vol. 64Google Scholar
  47. 47.
    D.W. Brown, V.M. Kenkre, in Electronic Structure and Properties of Hydrogen in Metals, edited by P. Jena, C. Satterthwaite (Plenum, New York, 1983), p. 177Google Scholar
  48. 48.
    V.M. Kenkre, D.W. Brown, Phys. Rev. B 31, 2479 (1985)ADSCrossRefGoogle Scholar
  49. 49.
    D.W. Brown, V.M. Kenkre, J. Phys. Chem. Solids 46, 579 (1985)ADSCrossRefGoogle Scholar
  50. 50.
    D.W. Brown, V.M. Kenkre, J. Phys. Chem. Solids 47, 289 (1986)ADSCrossRefGoogle Scholar
  51. 51.
    D.W. Brown, V.M. Kenkre, J. Phys. Chem. Solids 35, 456 (1987)Google Scholar
  52. 52.
    Y. Jung, E. Barkai, R. Silbey, Chem. Phys. 284, 181 (2002)ADSCrossRefGoogle Scholar
  53. 53.
    P. Hänggi, H. Thomas, Z. Phys. B 26, 85 (1977)ADSMathSciNetCrossRefGoogle Scholar
  54. 54.
    V.M. Kenkre, F. Sevilla, in Contributions to Mathematical Physics: a Tribute to Gerard G. Emch, edited by T.S. Ali, K.B. Sinha (Hindustani Book Agency, New Delhi, 2007), p. 147Google Scholar
  55. 55.
    R. Balescu, Physica 27, 693 (1961)ADSMathSciNetCrossRefGoogle Scholar
  56. 56.
    R.J. Swenson, Physica 29, 1174 (1963)ADSMathSciNetCrossRefGoogle Scholar
  57. 57.
    L. van Hove, Physica 23, 441 (1957)ADSMathSciNetCrossRefGoogle Scholar
  58. 58.
    I. Prigogine, P. Résibois, Physica 27, 629 (1961)ADSMathSciNetCrossRefGoogle Scholar
  59. 59.
    E.W. Montroll, in Fundamental Problems in Statistical Mechanics, compiled by E.G.D. Cohen (North-Holland Publishing Co, Amsterdam, 1962), p. 230Google Scholar
  60. 60.
    R.J. Swenson, J. Math. Phys. 3, 1017 (1962)ADSMathSciNetCrossRefGoogle Scholar
  61. 61.
    P. Résibois, Physica 29, 721 (1963)ADSMathSciNetCrossRefGoogle Scholar
  62. 62.
    R. Zwanzig, Physica 30, 1109 (1964)ADSMathSciNetCrossRefGoogle Scholar
  63. 63.
    V. Čápek, Czech. J. Phys. B 34, 1246 (1984)ADSCrossRefGoogle Scholar
  64. 64.
    A. Dechant et al., Phys. Rev. X 4, 011022 (2014)Google Scholar
  65. 65.
    S. Burova, R. Metzlerb, E. Barkai, Proc. Natl. Acad. Sci. USA 107, 13228 (2010)ADSMathSciNetCrossRefGoogle Scholar
  66. 66.
    P. Allegrini et al., Phys. Rev. E 68, 056123 (2003)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Matthew Chase
    • 1
  • Tom J. McKetterick
    • 2
    • 3
  • Luca Giuggioli
    • 2
    • 3
    • 4
  • V. M. Kenkre
    • 1
  1. 1.Consortium of the Americas for Interdisciplinary Science and the Department of Physics and Astronomy, University of New MexicoAlbuquerqueUSA
  2. 2.Bristol Centre for Complexity Sciences, University of BristolBristolUK
  3. 3.Department of Engineering MathematicsUniversity of BristolBristolUK
  4. 4.School of Biological Sciences, University of BristolBristolUK

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