Fractional kinetic model for granular compaction

  • S. Živković
  • Z. M. Jakšić
  • J. R. Šćepanović
  • I. Lončarević
  • Lj. Budinski-Petković
  • S. B. Vrhovac
Regular Article

Abstract

We present an approach to granular compaction based on subordination of stochastic processes. In order to imitate, in a very simplified way, the compaction dynamics of granular material under tapping, we impose that particles switch stochastically between the two possible orientational states characterizing the average volumes of the grain in the presence of other grains. The main physical idea of our approach is that the interaction of grains with their environment is taken into account with the aid of the temporal subordination. Accordingly, we assume that the time intervals between the consecutive grain’s reorientations are governed by a certain waiting-time distribution ψ(t). It is demonstrated how the presence of the trapping events leads to the macroscopic observation of slow compaction dynamics, described by an exact fractional kinetic equation. We also perform numerical simulations to examine our analytical result. In addition, we reproduce the memory effects numerically by considering the response of the system to the abrupt change in the external excitation.

Keywords

Statistical and Nonlinear Physics 

References

  1. 1.
    P. Richard, M. Nicodemi, R. Delannay, P. Ribière, D. Bideau, Nat. Mater. 4, 121 (2005)CrossRefADSGoogle Scholar
  2. 2.
    J.B. Knight, C.G. Fandrich, C.N. Lau, H.M. Jaeger, S.R. Nagel, Phys. Rev. E 51, 3957 (1995)CrossRefADSGoogle Scholar
  3. 3.
    F.X. Villarruel, B.E. Lauderdale, D.M. Mueth, H.M. Jaeger, Phys. Rev. E 61, 6914 (2000)CrossRefADSGoogle Scholar
  4. 4.
    P. Philippe, D. Bideau, Europhys. Lett. 60, 677 (2002)CrossRefADSGoogle Scholar
  5. 5.
    P. Ribière, P. Richard, D. Bideau, R. Delannay, Eur. Phys. J. E 16, 415 (2005)CrossRefGoogle Scholar
  6. 6.
    P. Ribière, P. Richard, P. Philippe, D. Bideau, R. Delannay, Eur. Phys. J. E 22, 249 (2007)CrossRefGoogle Scholar
  7. 7.
    G. Lumay, N. Vandewalle, Phys. Rev. Lett. 95, 028002 (2005)CrossRefADSGoogle Scholar
  8. 8.
    G. Lumay, N. Vandewalle, Phys. Rev. E 74, 021301 (2006)CrossRefADSGoogle Scholar
  9. 9.
    M. Nicolas, P. Duru, O. Pouliquen, Eur. Phys. J. E 3, 309 (2000)CrossRefGoogle Scholar
  10. 10.
    O. Pouliquen, M. Belzons, M. Nicolas, Phys. Rev. Lett. 91, 014301 (2003)CrossRefADSGoogle Scholar
  11. 11.
    D. Arsenović, S.B. Vrhovac, Z.M. Jakšić, L. Budinski-Petković, A. Belić, Phys. Rev. E 74, 061302 (2006)CrossRefADSGoogle Scholar
  12. 12.
    S. Živković, Z.M. Jakšić, D. Arsenović, L. Budinski-Petković, S.B. Vrhovac, Granular Matter 13, 493 (2011)CrossRefGoogle Scholar
  13. 13.
    R. Hilfer, J. Non-Cryst. Solids 305, 122 (2002)CrossRefADSGoogle Scholar
  14. 14.
    Lj. Budinski-Petković, M. Petković, Z.M. Jakšić, S.B. Vrhovac, Phys. Rev. E 72, 046118 (2005)CrossRefADSGoogle Scholar
  15. 15.
    J. Talbot, G. Tarjus, P. Viot, Phys. Rev. E 61, 5429 (2000)CrossRefADSGoogle Scholar
  16. 16.
    G. Tarjus, P. Viot, Phys. Rev. E 69, 011307 (2004)CrossRefADSGoogle Scholar
  17. 17.
    Lj. Budinski-Petković, S.B. Vrhovac, Eur. Phys. J. E 16, 89 (2005)CrossRefGoogle Scholar
  18. 18.
    W.L. Vargas, J.J. McCarthy, Phys. Rev. E 76, 041301 (2007)CrossRefADSGoogle Scholar
  19. 19.
    T. Divoux, H. Gayvallet, J.-C. Géminard, Phys. Rev. Lett. 101, 148303 (2008)CrossRefADSGoogle Scholar
  20. 20.
    N. Mueggenburg, Phys. Rev. E 85, 041305 (2012)CrossRefADSGoogle Scholar
  21. 21.
    A.J. Kolan, E.R. Nowak, A.V. Tkachenko, Phys. Rev. E 59, 3094 (1999)CrossRefADSGoogle Scholar
  22. 22.
    J. Talbot, G. Tarjus, P. Viot, Eur. Phys. J. E 5, 445 (2001)CrossRefGoogle Scholar
  23. 23.
    M. Nicodemi, A. Coniglio, Phys. Rev. Lett. 82, 916 (1999)CrossRefADSGoogle Scholar
  24. 24.
    A. Barrat, V. Loreto, J. Phys. A, 33 4401(2000)CrossRefMATHMathSciNetADSGoogle Scholar
  25. 25.
    A. Barrat, V. Loreto, Europhys. Lett. 53, 297 (2001)CrossRefADSGoogle Scholar
  26. 26.
    A. Fierro, M. Nicodemi, A. Coniglio, Phys. Rev. E 66, 061301 (2002)CrossRefADSGoogle Scholar
  27. 27.
    P.F. Stadler, J.M. Luck, A. Mehta, Eurohys. Lett. 57, 46 (2001)CrossRefADSGoogle Scholar
  28. 28.
    J.J. Brey, A. Prados, B. Sanchez-Rey, Phys. Rev. E 60, 5685 (1999)CrossRefADSGoogle Scholar
  29. 29.
    J.J. Brey, A. Prados, Phys. Rev. E 63, 061301 (2001)CrossRefADSGoogle Scholar
  30. 30.
    A. Prados, J.J. Brey, Phys. Rev. E 66, 041308 (2002)CrossRefADSGoogle Scholar
  31. 31.
    J.J. Brey, A. Prados, Phys. Rev. E 68, 051302 (2003)CrossRefADSGoogle Scholar
  32. 32.
    S.F. Edwards, D.V. Grinev, Phys. Rev. E 58, 4758 (1998)CrossRefADSGoogle Scholar
  33. 33.
    C. Josserand, A. Tkachenko, D.M. Mueth, H.M. Jaeger, Phys. Rev. Lett. 85, 3632 (2000)CrossRefADSGoogle Scholar
  34. 34.
    L.A. Pugnaloni, G.C. Barker, Physica A 337, 428 (2004)CrossRefADSGoogle Scholar
  35. 35.
    R. Arévalo, D. Maza, L.A. Pugnaloni, Phys. Rev. E 74, 021303 (2006)CrossRefADSGoogle Scholar
  36. 36.
    L.A. Pugnaloni, M.G. Valluzzi, L.G. Valluzzi, Phys. Rev. E 73, 051302 (2006)CrossRefADSGoogle Scholar
  37. 37.
    G. Marty, O. Dauchot, Phys. Rev. Lett. 94, 015701 (2005)CrossRefADSGoogle Scholar
  38. 38.
    R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)CrossRefMATHMathSciNetADSGoogle Scholar
  39. 39.
    A. Mehta, G.C. Barker, J.M. Luck, J. Stat. Mech.: Theor. Exp. 2004, P10014 (2004)CrossRefGoogle Scholar
  40. 40.
    P. Ribière, P. Richard, R. Delannay, D. Bideau, Phys. Rev. Lett. 95, 268001 (2005)CrossRefADSGoogle Scholar
  41. 41.
    A. Janicki, A. Weron, Stat. Sci. 9, 109 (1994)CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    M.M. Meerschaert, H.-P. Scheffler, J. Appl. Probab. 41, 623 (2004)CrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    F. Mainardi, Chaos Solitons Fractals 7, 1461 (1996)CrossRefMATHMathSciNetADSGoogle Scholar
  44. 44.
    A.A. Stanislavsky, Phys. Rev. E 67, 021111 (2003)CrossRefMathSciNetADSGoogle Scholar
  45. 45.
    A.A. Stanislavsky, Acta Physica Polonica B 34, 3649 (2003)ADSGoogle Scholar
  46. 46.
    F. Mainardi, R. Gorenflo, J. Comput. Appl. Math. 118, 283 (2000)CrossRefMATHMathSciNetADSGoogle Scholar
  47. 47.
    R.K. Saxena, A.M. Mathai, H.J. Haubold, Physica A 344, 657 (2004)CrossRefMathSciNetADSGoogle Scholar
  48. 48.
    P. Philippe, D. Bideau, Phys. Rev. Lett. 91, 104302 (2003)CrossRefADSGoogle Scholar
  49. 49.
    N. Gravish, S.V. Franklin, D.L. Hu, D.I. Goldman, Phys. Rev. Lett. 108, 208001 (2012)CrossRefADSGoogle Scholar
  50. 50.
    E. Heinsalu, M. Patriarca, I. Goychuk, G. Schmid, P. Hänggi, Phys. Rev. E 73, 046133 (2006)CrossRefADSGoogle Scholar
  51. 51.
    E. Heinsalu, M. Patriarca, I. Goychuk, P. Hänggi, J. Phys.: Condens. Matter 19, 065114 (2007)ADSGoogle Scholar
  52. 52.
    R. Hilfer, L. Anton, Phys. Rev. E 51, R848 (1995)CrossRefADSGoogle Scholar
  53. 53.
    D. Fulger, E. Scalas, G. Germano, Phys. Rev. E 77, 021122 (2008)CrossRefADSGoogle Scholar
  54. 54.
    T.J. Kozubowski, S.T. Rachev, Int. J. Comput. Numer. Anal. Appl. 1, 177 (1999)MATHMathSciNetGoogle Scholar
  55. 55.
    E.R. Nowak, J.B. Knight, M.L. Povinelli, H.M. Jaeger, S.R. Nagel, Powder Technol. 94, 79 (1997)CrossRefGoogle Scholar
  56. 56.
    E.R. Nowak, J.B. Knight, E. Ben-Naim, H.M. Jaeger, S.R. Nagel, Phys. Rev. E 57, 1971 (1998)CrossRefADSGoogle Scholar
  57. 57.
    I.M. Sokolov, A. Blumen, J. Klafter, Europhys. Lett. 56, 175 (2001)CrossRefADSGoogle Scholar
  58. 58.
    I.M. Sokolov, A. Blumen, J. Klafter, Physica A 302, 268 (2001)CrossRefMATHADSGoogle Scholar
  59. 59.
    I.M. Sokolov, Phys. Rev. E 73, 067102 (2006)Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • S. Živković
    • 1
  • Z. M. Jakšić
    • 1
  • J. R. Šćepanović
    • 1
  • I. Lončarević
    • 2
  • Lj. Budinski-Petković
    • 2
  • S. B. Vrhovac
    • 1
  1. 1.Institute of Physics BelgradeUniversity of BelgradeZemun, BelgradeSerbia
  2. 2.Faculty of EngineeringNovi SadSerbia

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